Module

CyclotomicAbstractFixedFieldArtin.lean

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtin

107 declarations indexed from sourceView source ↗

Module documentation

Cyclotomic Artin coordinates over abstract fixed fields

The cyclotomic degree datum on the rational absolute Galois group has an actual maximal-unramified field over every finite abstract fixed field. This file identifies its genuine Galois group with Multiplicative ZHat, using the normalized degree map, and supplies the abelian Galois structure needed by the actual infinite global Artin homomorphism.

These constructions are the source side of the comparison between abstract finite reciprocity and the chosen local-factor product. No reciprocity comparison is assumed in their definitions.

Imports3

Imported by2

instance

cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra

L41 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra

Documentation

Keep this module on the rational algebra structures used by the cyclotomic fixed-field API. Generic intermediate-field instances are propositionally equal here but not definitionally interchangeable.

Source context 6 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ)

instance

cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra

L46 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra

Source context 7 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField

theorem

rationalCyclotomicFieldInertia_le

L53 ↗

GlobalClassFieldTheory.Reciprocity.rationalCyclotomicFieldInertia_le

Documentation

Cyclotomic field inertia is contained in the original abstract field subgroup.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · statement

theorem rationalCyclotomicFieldInertia_le
    (H : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    (rationalCyclotomicDegreeData.fieldInertia H).toSubgroup ≤
      H.toSubgroup

theorem

extensionSubgroup_rationalCyclotomicFieldInertia

L63 ↗

GlobalClassFieldTheory.Reciprocity.extensionSubgroup_rationalCyclotomicFieldInertia

Documentation

Viewing cyclotomic field inertia inside its ambient field subgroup gives exactly the kernel of normalized degree.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · statement

theorem extensionSubgroup_rationalCyclotomicFieldInertia
    (H : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    CyclicCohomology.extensionSubgroup H
        (rationalCyclotomicDegreeData.fieldInertia H)
        (rationalCyclotomicFieldInertia_le H) =
      rationalCyclotomicDegreeData.fieldInertiaWithin H

def

abstractFixedFieldCyclotomicGalEquivZHat

L80 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat

Documentation

The actual Galois group of the cyclotomic maximal-unramified extension of an abstract fixed field, in its normalized ZHat coordinate.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

def abstractFixedFieldCyclotomicGalEquivZHat
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    Gal(
        LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI /
        LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field) ≃*
      Multiplicative ZHat := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let qField :
      H.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI ≃*
        Gal(
          LocalClassFieldTheory.abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) hI /
          LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
      ℚ (SeparableClosure ℚ) H.field
      (rationalCyclotomicDegreeData.fieldInertia H.field)
      hI hnormal
  let qInertia :
      H.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI ≃*
        H.field.toSubgroup ⧸
          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
    QuotientGroup.quotientMulEquivOfEq
      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
  exact
    qField.symm.trans
      (qInertia.trans
        (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
          (H.toFiniteResidueAbstractField
            rationalCyclotomicDegreeData)))

theorem

abstractFixedFieldCyclotomicGalEquivZHat_extensionClass

L132 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat_extensionClass

Documentation

On an absolute-Galois representative fixing the lower field, the actual cyclotomic Galois coordinate is its normalized degree.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

theorem abstractFixedFieldCyclotomicGalEquivZHat_extensionClass
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (σ : H.field.toSubgroup) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let hnormal :
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI).Normal := by
      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
      infer_instance
    abstractFixedFieldCyclotomicGalEquivZHat H
        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
          ℚ (SeparableClosure ℚ) H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI hnormal
          (QuotientGroup.mk σ)) =
      rationalCyclotomicDegreeData.normalizedDegree
        (H.toFiniteResidueAbstractField
          rationalCyclotomicDegreeData) σ := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let qField :=
    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
      ℚ (SeparableClosure ℚ) H.field
      (rationalCyclotomicDegreeData.fieldInertia H.field)
      hI hnormal
  change
    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
      (H.toFiniteResidueAbstractField
        rationalCyclotomicDegreeData))
      (QuotientGroup.quotientMulEquivOfEq
        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
        (qField.symm (qField (QuotientGroup.mk σ)))) =
      rationalCyclotomicDegreeData.normalizedDegree
        (H.toFiniteResidueAbstractField
          rationalCyclotomicDegreeData) σ
  rw [qField.symm_apply_apply]
  rfl

theorem

abstractFixedFieldCyclotomicGalEquivZHat_quotientClass

L182 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat_quotientClass

Documentation

Quotient-level evaluation of the actual cyclotomic Galois coordinate.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

theorem abstractFixedFieldCyclotomicGalEquivZHat_quotientClass
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (q :
      H.field.toSubgroup ⧸
        rationalCyclotomicDegreeData.fieldInertiaWithin H.field) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let hnormal :
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI).Normal := by
      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
      infer_instance
    let qInertia :
        H.field.toSubgroup ⧸
            CyclicCohomology.extensionSubgroup H.field
              (rationalCyclotomicDegreeData.fieldInertia H.field)
              hI ≃*
          H.field.toSubgroup ⧸
            rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
      QuotientGroup.quotientMulEquivOfEq
        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
    abstractFixedFieldCyclotomicGalEquivZHat H
        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
          ℚ (SeparableClosure ℚ) H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI hnormal
          (qInertia.symm q)) =
      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
        (H.toFiniteResidueAbstractField
          rationalCyclotomicDegreeData) q := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let qField :=
    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
      ℚ (SeparableClosure ℚ) H.field
      (rationalCyclotomicDegreeData.fieldInertia H.field)
      hI hnormal
  let qInertia :
      H.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI ≃*
        H.field.toSubgroup ⧸
          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
    QuotientGroup.quotientMulEquivOfEq
      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
  change
    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
      (H.toFiniteResidueAbstractField
        rationalCyclotomicDegreeData))
      (qInertia
        (qField.symm
          (qField (qInertia.symm q)))) =
      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
        (H.toFiniteResidueAbstractField
          rationalCyclotomicDegreeData) q
  rw [qField.symm_apply_apply, qInertia.apply_symm_apply]

theorem

abstractFixedFieldCyclotomic_isAbelianGalois

L250 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomic_isAbelianGalois

Documentation

The extension fixed by cyclotomic field inertia is an actual abelian Galois extension of the lower abstract fixed field.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

theorem abstractFixedFieldCyclotomic_isAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    IsAbelianGalois
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let : IsGalois
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) :=
    LocalClassFieldTheory.abstractRelativeFixedField_isGalois
      ℚ (SeparableClosure ℚ) H.field
      (rationalCyclotomicDegreeData.fieldInertia H.field)
      hI hnormal
  let e :=
    abstractFixedFieldCyclotomicGalEquivZHat H
  exact
    { is_comm.comm := by
        intro σ τ
        apply e.injective
        simpa only [map_mul] using mul_comm (e σ) (e τ) }

def

rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum

L287 ↗

GlobalClassFieldTheory.Reciprocity.rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum

Documentation

The rational cyclotomic ZHat-field embedded in the actual maximal-unramified compositum of an abstract fixed field.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

def
    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    rationalCyclotomicZHatField →ₐ[ℚ]
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let J :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ)
      (rationalCyclotomicDegreeData.fieldInertia H.field)
  have hTJ :
      rationalCyclotomicZHatField ≤ J := by
    change
      rationalCyclotomicZHatField ≤
        LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicDegreeData.fieldInertia H.field)
    rw [
      rationalCyclotomicDegreeData_fixedField_fieldInertia]
    exact le_sup_right
  exact IntermediateField.inclusion hTJ

instance

abstractFixedFieldCyclotomicCompositum_algebra

L314 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicCompositum_algebra

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

instance
    abstractFixedFieldCyclotomicCompositum_algebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    Algebra rationalCyclotomicZHatField
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) :=
  ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
    H).toRingHom).toAlgebra

instance

abstractFixedFieldCyclotomicCompositum_scalarTower

L326 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicCompositum_scalarTower

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

instance abstractFixedFieldCyclotomicCompositum_scalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    IsScalarTower ℚ rationalCyclotomicZHatField
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) :=
  IsScalarTower.of_algebraMap_eq'
    ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
      H).comp_algebraMap).symm

instance

abstractFixedFieldCyclotomicCompositum_baseScalarTower

L338 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicCompositum_baseScalarTower

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

instance abstractFixedFieldCyclotomicCompositum_baseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    IsScalarTower ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  rfl

def

abstractFixedFieldCyclotomicRestriction

L357 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction

Documentation

Restriction from the actual maximal-unramified compositum of an abstract fixed field to the rational cyclotomic factor.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · full declaration

def abstractFixedFieldCyclotomicRestriction
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    Gal(
        LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI /
        LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field) →*
      Gal(rationalCyclotomicZHatField / ℚ) := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  exact
    IntermediateField.restrictRestrictAlgEquivMapHom
      ℚ rationalCyclotomicZHatField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI)

instance

cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal

L381 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal

Documentation

On a quotient representative, cyclotomic restriction of the actual relative automorphism is ordinary restriction of the same ambient absolute-Galois automorphism.

Source context 8 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField

theorem

abstractFixedFieldCyclotomicRestriction_extensionClass

L387 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction_extensionClass

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · full declaration

theorem abstractFixedFieldCyclotomicRestriction_extensionClass
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (σ : H.field.toSubgroup) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let hnormal :
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI).Normal := by
      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
      infer_instance
    abstractFixedFieldCyclotomicRestriction H
        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
          ℚ (SeparableClosure ℚ) H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI hnormal
          (QuotientGroup.mk σ)) =
      AlgEquiv.restrictNormalHom
        rationalCyclotomicZHatField σ.1 := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let τ :=
    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
      ℚ (SeparableClosure ℚ) H.field
      (rationalCyclotomicDegreeData.fieldInertia H.field)
      hI hnormal
      (QuotientGroup.mk'
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field) hI)
        σ)
  apply AlgEquiv.ext
  intro x
  apply Subtype.ext
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) hI
  have hrestrict :
      algebraMap rationalCyclotomicZHatField U
          ((abstractFixedFieldCyclotomicRestriction H τ) x) =
        τ (algebraMap rationalCyclotomicZHatField U x) := by
    change
      algebraMap rationalCyclotomicZHatField U
          ((AlgEquiv.restrictNormal
            (MulSemiringAction.toAlgEquiv ℚ U τ)
            rationalCyclotomicZHatField) x) =
        (MulSemiringAction.toAlgEquiv ℚ U τ)
          (algebraMap rationalCyclotomicZHatField U x)
    exact
      AlgEquiv.restrictNormal_commutes
        (MulSemiringAction.toAlgEquiv ℚ U τ)
        rationalCyclotomicZHatField x
  have halgebraMap_eq_embedding
      (z : rationalCyclotomicZHatField) :
      algebraMap rationalCyclotomicZHatField U z =
        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
          H z : U) := by
    rfl
  have hembedding_coe
      (z : rationalCyclotomicZHatField) :
      (((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
          H z : U) : SeparableClosure ℚ)) =
        (z : SeparableClosure ℚ) := by
    rfl
  have halgebraMap_coe
      (z : rationalCyclotomicZHatField) :
      ((algebraMap rationalCyclotomicZHatField U z : U) :
          SeparableClosure ℚ) =
        (z : SeparableClosure ℚ) := by
    rw [halgebraMap_eq_embedding]
    exact hembedding_coe z
  have hambient :=
    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
        ℚ (SeparableClosure ℚ) H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI hnormal σ
        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
          H x)
  calc
    (((abstractFixedFieldCyclotomicRestriction H τ) x :
        rationalCyclotomicZHatField) :
        SeparableClosure ℚ) =
        ((τ
          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
            H x) :
          LocalClassFieldTheory.abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) hI) :
          SeparableClosure ℚ) := by
      calc
        (((abstractFixedFieldCyclotomicRestriction H τ) x :
            rationalCyclotomicZHatField) : SeparableClosure ℚ) =
            ((algebraMap rationalCyclotomicZHatField U
              ((abstractFixedFieldCyclotomicRestriction H τ) x) : U) :
              SeparableClosure ℚ) :=
          (halgebraMap_coe
            ((abstractFixedFieldCyclotomicRestriction H τ) x)).symm
        _ = ((τ (algebraMap rationalCyclotomicZHatField U x) : U) :
            SeparableClosure ℚ) :=
          congrArg (fun y : U => (y : SeparableClosure ℚ)) hrestrict
        _ = ((τ
            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
              H x) : U) : SeparableClosure ℚ) := by
          rw [halgebraMap_eq_embedding]
    _ = σ.1 (x : SeparableClosure ℚ) := by
      calc
        ((τ
            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
              H x) : U) : SeparableClosure ℚ) =
            σ.1
              ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
                H x : U) : SeparableClosure ℚ) := by
          simpa only [τ, U] using hambient.symm
        _ = σ.1 (x : SeparableClosure ℚ) := by
          rw [hembedding_coe]
    _ =
        (((AlgEquiv.restrictNormalHom
          rationalCyclotomicZHatField σ.1) x :
          rationalCyclotomicZHatField) :
          SeparableClosure ℚ) := by
      exact
        (AlgEquiv.restrictNormal_commutes
          σ.1 rationalCyclotomicZHatField x).symm

theorem

abstractFixedFieldCyclotomicRestriction_coordinate

L518 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction_coordinate

Documentation

Raw cyclotomic restriction is residue-degree multiplication of the normalized actual Galois coordinate.

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

theorem
    abstractFixedFieldCyclotomicRestriction_coordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (τ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    Multiplicative.toAdd
        (rationalCyclotomicZHatFieldGalEquivZHat
          (abstractFixedFieldCyclotomicRestriction H τ)) =
      (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
        Multiplicative.toAdd
          (abstractFixedFieldCyclotomicGalEquivZHat H τ)

def

abstractFixedFieldCyclotomicFiniteCompositum

L559 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum

Documentation

The canonical compositum of an abstract fixed field with a finite layer of the rational cyclotomic ZHat-extension.

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

def abstractFixedFieldCyclotomicFiniteCompositum
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IntermediateField ℚ (SeparableClosure ℚ)

instance

abstractFixedFieldCyclotomicFiniteCompositum_finiteDimensional

L570 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_finiteDimensional

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_finiteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    FiniteDimensional ℚ
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

instance

abstractFixedFieldCyclotomicFiniteCompositum_numberField

L592 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_numberField

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

def

abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding

L606 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding

Documentation

The lower abstract fixed field embedded into its finite cyclotomic compositum.

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

def
    abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field →ₐ[ℚ]
      abstractFixedFieldCyclotomicFiniteCompositum H E

instance

abstractFixedFieldCyclotomic_numberField

L626 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomic_numberField

Documentation

The fixed field attached to a finite abstract field is a number field. Keeping this as the single file-local instance makes it available while later theorem binders are elaborated.

Source context 9 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal

Lean source · statement

instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

def

abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding

L645 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding

Documentation

A finite rational cyclotomic layer embedded into its compositum with the abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

def
    abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    E →ₐ[ℚ]
      abstractFixedFieldCyclotomicFiniteCompositum H E

instance

abstractFixedFieldCyclotomicFiniteCompositum_baseAlgebra

L661 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_baseAlgebra

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_baseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

instance

abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra

L676 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

instance

abstractFixedFieldCyclotomicFiniteCompositum_layerSMul

L691 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_layerSMul

Documentation

The finite-layer action induced by its explicit embedding into the finite compositum.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_layerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

instance

abstractFixedFieldCyclotomicFiniteCompositum_baseScalarTower

L704 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_baseScalarTower

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_baseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

instance

abstractFixedFieldCyclotomicFiniteCompositum_layerScalarTower

L719 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositum_layerScalarTower

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteCompositum_layerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (abstractFixedFieldCyclotomicFiniteCompositum H E)

def

abstractFixedFieldCyclotomicFiniteCompositumInclusion

L734 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositumInclusion

Documentation

Inclusion of the finite cyclotomic compositum into the actual maximal-unramified compositum.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

def
    abstractFixedFieldCyclotomicFiniteCompositumInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[ℚ]
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI := by
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let J :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ)
      (rationalCyclotomicDegreeData.fieldInertia H.field)
  have hle :
      abstractFixedFieldCyclotomicFiniteCompositum H E ≤ J := by
    dsimp only [J]
    rw [
      rationalCyclotomicDegreeData_fixedField_fieldInertia H.field]
    exact
      sup_le_sup le_rfl
        (IntermediateField.lift_le E.toIntermediateField)
  exact IntermediateField.inclusion hle

def

abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase

L764 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase

Documentation

The same finite-compositum inclusion over the lower abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

def
    abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field]
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI := by
  let f :=
    abstractFixedFieldCyclotomicFiniteCompositumInclusion H E
  exact
    { f.toRingHom with
      commutes' := by
        intro x
        rfl }

def

abstractFixedFieldCyclotomicFiniteLayer

L788 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer

Documentation

The finite cyclotomic compositum as an intermediate field of the actual maximal-unramified extension.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

def abstractFixedFieldCyclotomicFiniteLayer
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    IntermediateField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) :=
  (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase H E).fieldRange

instance

abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra

L805 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra

Documentation

The base algebra on the finite field range, obtained from the explicit base embedding followed by the field-range equivalence.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_baseSMul

L827 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_baseSMul

Documentation

The scalar action belonging to the canonical base algebra on the finite field range.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_baseSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_baseModule

L843 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_baseModule

Documentation

The module structure belonging to the canonical base algebra on the finite field range.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_baseModule
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Module
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

def

abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer

L863 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer

Documentation

The field-range equivalence rebuilt over the explicit base algebras. Its underlying ring equivalence is the canonical one.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

def
    abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    abstractFixedFieldCyclotomicFiniteCompositum H E ≃ₐ[
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field]
      abstractFixedFieldCyclotomicFiniteLayer H E

instance

abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional

L881 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    FiniteDimensional
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_numberField

L895 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_numberField

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois

L909 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_baseScalarTower

L937 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_baseScalarTower

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_baseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)

def

abstractFixedFieldCyclotomicFiniteLayerEmbedding

L975 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayerEmbedding

Documentation

The finite rational layer embedded into its corresponding intermediate field over the abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

def
    abstractFixedFieldCyclotomicFiniteLayerEmbedding
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    E →ₐ[ℚ]
      abstractFixedFieldCyclotomicFiniteLayer H E

instance

abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra

L991 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_layerSMul

L1006 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_layerSMul

Documentation

The finite-layer action on its actual image in the relative fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_layerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (abstractFixedFieldCyclotomicFiniteLayer H E)

instance

abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower

L1018 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (abstractFixedFieldCyclotomicFiniteLayer H E)

def

abstractFixedFieldCyclotomicFiniteGaloisLayer

L1034 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteGaloisLayer

Documentation

The finite cyclotomic layer as an object of the finite-Galois inverse system of the actual maximal-unramified extension.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

def
    abstractFixedFieldCyclotomicFiniteGaloisLayer
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    FiniteGaloisIntermediateField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ)
        (rationalCyclotomicFieldInertia_le H.field)) where
  toIntermediateField :=
    abstractFixedFieldCyclotomicFiniteLayer H E
  finiteDimensional :=
    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional H E
  isGalois :=
    (abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E).toIsGalois

theorem

abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra_eq_algebra'

L1056 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra_eq_algebra'

Documentation

The explicit base algebra on a finite cyclotomic layer is the canonical intermediate-field inclusion used by the finite Galois inverse system.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra_eq_algebra'
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E =
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'

instance

abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower

L1071 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower

Documentation

The finite Galois layer uses its canonical inclusion into the full relative fixed field for the upper scalar action.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

instance
    abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).toIntermediateField
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ)
        (rationalCyclotomicFieldInertia_le H.field))

def

abstractFixedFieldCyclotomicFiniteLayerInclusion

L1104 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayerInclusion

Documentation

The canonical inclusion of the finite cyclotomic layer into the full abstract-fixed-field compositum.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

def
    abstractFixedFieldCyclotomicFiniteLayerInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    abstractFixedFieldCyclotomicFiniteLayer H E →ₐ[
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field]
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ)
        (rationalCyclotomicFieldInertia_le H.field)

theorem

abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion

L1122 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion

Documentation

The two embeddings of a finite rational cyclotomic layer into the full abstract-fixed-field compositum agree.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    (z : E) :
    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z) =
      rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
        (z : rationalCyclotomicZHatField)

theorem

restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply

L1138 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply

Documentation

Restriction to E commutes pointwise with the restriction from the full rational cyclotomic tower.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal ℚ E]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (z : E) :
    ((AlgEquiv.restrictNormalHom E
      (abstractFixedFieldCyclotomicRestriction H σ)) z :
        rationalCyclotomicZHatField) =
      (abstractFixedFieldCyclotomicRestriction H σ)
        (z : rationalCyclotomicZHatField)

theorem

abstractFixedFieldCyclotomicRestriction_embedding_apply

L1163 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction_embedding_apply

Documentation

The raw cyclotomic restriction commutes with the canonical embedding of the full rational cyclotomic tower.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    abstractFixedFieldCyclotomicRestriction_embedding_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (z : rationalCyclotomicZHatField) :
    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
        (abstractFixedFieldCyclotomicRestriction H σ z) =
      σ
        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
          H z)

theorem

restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply

L1189 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply

Documentation

Restriction to the finite compositum layer commutes with its canonical inclusion into the full compositum.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (z : abstractFixedFieldCyclotomicFiniteLayer H E) :
    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
        (AlgEquiv.restrictNormalHom
          (abstractFixedFieldCyclotomicFiniteLayer H E) σ z) =
      σ (abstractFixedFieldCyclotomicFiniteLayerInclusion H E z)

theorem

abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply

L1225 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply

Documentation

Restricting the finite-compositum action further to E commutes with the explicit embedding of E into that finite layer.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal ℚ E]
    [Normal
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (z : E) :
    abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
        (IntermediateField.restrictRestrictAlgEquivMapHom
          ℚ E
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field)
          (abstractFixedFieldCyclotomicFiniteLayer H E)
          (AlgEquiv.restrictNormalHom
            (abstractFixedFieldCyclotomicFiniteLayer H E) σ) z) =
      (AlgEquiv.restrictNormalHom
        (abstractFixedFieldCyclotomicFiniteLayer H E) σ)
        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z)

theorem

restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded

L1270 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded

Documentation

The left finite-level restriction, after both canonical embeddings into the full compositum, is the action of σ on the cyclotomic embedding.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal ℚ E]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (x : E) :
    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
          (AlgEquiv.restrictNormalHom E
            (abstractFixedFieldCyclotomicRestriction H σ) x)) =
      σ
        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
          (x : rationalCyclotomicZHatField))

theorem

restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded

L1316 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded

Documentation

The right finite-level restriction, after both canonical embeddings into the full compositum, is the same action of σ.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal ℚ E]
    [Normal
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (x : E) :
    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
          (IntermediateField.restrictRestrictAlgEquivMapHom
            ℚ E
            (LocalClassFieldTheory.abstractFixedField
              ℚ (SeparableClosure ℚ) H.field)
            (abstractFixedFieldCyclotomicFiniteLayer H E)
            (AlgEquiv.restrictNormalHom
              (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x)) =
      σ
        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
          (x : rationalCyclotomicZHatField))

theorem

restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise

L1376 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise

Documentation

Pointwise form of finite-layer restriction compatibility.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    [Normal ℚ E]
    [Normal
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (abstractFixedFieldCyclotomicFiniteLayer H E)]
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field))
    (x : E) :
    AlgEquiv.restrictNormalHom E
        (abstractFixedFieldCyclotomicRestriction H σ) x =
      IntermediateField.restrictRestrictAlgEquivMapHom
          ℚ E
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field)
          (abstractFixedFieldCyclotomicFiniteLayer H E)
        (AlgEquiv.restrictNormalHom
          (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x

theorem

restrictNormalHom_abstractFixedFieldCyclotomicRestriction

L1415 ↗

GlobalClassFieldTheory.Reciprocity.restrictNormalHom_abstractFixedFieldCyclotomicRestriction

Documentation

Restricting through a finite layer commutes with restriction from the full abstract-fixed-field compositum.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    (σ :
      Gal(LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field)/LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    letI : Normal ℚ E := E.isGalois.to_normal
    AlgEquiv.restrictNormalHom E
        (abstractFixedFieldCyclotomicRestriction H σ) =
      IntermediateField.restrictRestrictAlgEquivMapHom
          ℚ E
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field)
          (abstractFixedFieldCyclotomicFiniteLayer H E)
        (AlgEquiv.restrictNormalHom
          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
          σ) := by
  let : Normal ℚ E := E.isGalois.to_normal
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ)
      (rationalCyclotomicFieldInertia_le H.field)
  let T := rationalCyclotomicZHatField
  let P : IntermediateField F U :=
    abstractFixedFieldCyclotomicFiniteLayer H E
  let : Algebra T U :=
    abstractFixedFieldCyclotomicCompositum_algebra H
  let : IsScalarTower ℚ T U :=
    abstractFixedFieldCyclotomicCompositum_scalarTower H
  let : Algebra E P :=
    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
  let : IsScalarTower ℚ E P :=
    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
  let : IsAbelianGalois F P := by
    change IsAbelianGalois F
      (abstractFixedFieldCyclotomicFiniteLayer H E)
    exact
      abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E
  let : Normal F P := IsGalois.to_normal
  apply AlgEquiv.ext
  intro x
  exact
    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
      H E σ x

theorem

cyclotomicAbstractFixedFieldArtin_eqTransThree

L1473 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtin_eqTransThree

Documentation

Opaque three-step equality composition used to keep large dependent finite-level coordinates out of endpoint proof normalization.

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers

Lean source · statement

theorem cyclotomicAbstractFixedFieldArtin_eqTransThree
    {α : Type} {a b c d : α}
    (hab : a = b) (hbc : b = c) (hcd : c = d) :
    a = d

abbrev

cyclotomicAbstractFixedFieldArtinCoordinateBase

L1479 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBase

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers

Lean source · statement

abbrev cyclotomicAbstractFixedFieldArtinCoordinateBase
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))

abbrev

cyclotomicAbstractFixedFieldArtinCoordinateRelative

L1485 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRelative

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers

Lean source · statement

abbrev cyclotomicAbstractFixedFieldArtinCoordinateRelative
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))

abbrev

cyclotomicAbstractFixedFieldArtinCoordinateLayer

L1492 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayer

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers

Lean source · statement

abbrev cyclotomicAbstractFixedFieldArtinCoordinateLayer
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IntermediateField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra

L1503 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra

Source context 11 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional

L1511 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional

Source context 12 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField

L1520 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField

Source context 13 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra

L1529 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra

Source context 14 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ)

instance

cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra

L1539 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra

Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower

L1548 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower

Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois

L1557 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois

Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateAlgebra

L1566 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateAlgebra

Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E

instance

cyclotomicAbstractFixedFieldArtinCoordinateNumberField

L1574 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateNumberField

Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E

instance

cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois

L1582 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois

Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E

instance

cyclotomicAbstractFixedFieldArtinCoordinateNormal

L1590 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateNormal

Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra

L1598 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra

Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra

L1610 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra

Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower

L1621 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower

Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField

L1655 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField

Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra

L1666 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra

Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul

L1677 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower

L1688 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra

L1699 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra

Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower

L1712 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower

Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)

instance

cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois

L1725 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois

Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E

Lean source · statement

instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)

def

cyclotomicAbstractFixedFieldArtinAbstractEndpoint

L1739 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinAbstractEndpoint

Documentation

The full abstract Artin symbol whose finite coordinates are compared below. Naming this endpoint keeps its relative fixed-field data opaque.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinAbstractEndpoint
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
    Gal(rationalCyclotomicZHatField / ℚ)

def

cyclotomicAbstractFixedFieldArtinRationalEndpoint

L1752 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinRationalEndpoint

Documentation

The rational norm Artin symbol serving as the other full endpoint.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinRationalEndpoint
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
    Gal(rationalCyclotomicZHatField / ℚ)

def

cyclotomicAbstractFixedFieldArtinCoordinateMapData

L1766 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateMapData

Documentation

The finite restriction map packaged together with its pointwise Artin naturality law. The map is inferred from the generic hom-level theorem, so no concrete instance tower is compared after the opaque boundary.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def cyclotomicAbstractFixedFieldArtinCoordinateMapData
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    {f : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
        Gal(E / ℚ) //
      f.comp
          (@globalArtinMonoidHomOfNumberField
            (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
            (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
            (inferInstance : Field
              (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
            (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
            (inferInstance : Field
              (abstractFixedFieldCyclotomicFiniteGaloisLayer H E))
            (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
            (cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois H E)
            (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)) =
        (globalArtinMonoidHom (K := ℚ) (L := E)).comp
          (IdeleGroup.norm ℚ
            (cyclotomicAbstractFixedFieldArtinCoordinateBase H))}

def

cyclotomicAbstractFixedFieldArtinCoordinateMap

L1835 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateMap

Documentation

The fixed restriction map from the relative finite layer to one rational cyclotomic coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def cyclotomicAbstractFixedFieldArtinCoordinateMap
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
      Gal(E / ℚ)

theorem

cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality

L1848 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality

Documentation

Naturality of the named coordinate map, kept at the hom level so later pointwise rewrites match the opaque map without unfolding its data package.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    (cyclotomicAbstractFixedFieldArtinCoordinateMap H E).comp
        (globalArtinMonoidHom
          (K := cyclotomicAbstractFixedFieldArtinCoordinateBase H)
          (L := abstractFixedFieldCyclotomicFiniteGaloisLayer H E)) =
      (globalArtinMonoidHom (K := ℚ) (L := E)).comp
        (IdeleGroup.norm ℚ
          (cyclotomicAbstractFixedFieldArtinCoordinateBase H))

theorem

cyclotomicAbstractFixedFieldArtinCoordinateMap_apply

L1865 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateMap_apply

Documentation

Pointwise identification of the named coordinate map with the concrete two-stage restriction used by the abstract fixed-field comparison.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)
    (σ : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer
      H E/cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
    cyclotomicAbstractFixedFieldArtinCoordinateMap H E σ =
      @IntermediateField.restrictRestrictAlgEquivMapHom
        ℚ E
        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
        (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
        inferInstance inferInstance inferInstance inferInstance
        (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
        (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
        (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
        (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
        (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)
        (cyclotomicAbstractFixedFieldArtinCoordinateNormal E) σ

def

cyclotomicAbstractFixedFieldArtinCoordinateBridgeData

L1892 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateBridgeData

Documentation

The relative projection, common rational finite value, and rational infinite projection, with both comparison steps packaged by the generic provider before this concrete tower becomes opaque.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinCoordinateBridgeData
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField)

def

cyclotomicAbstractFixedFieldArtinAbstractCoordinate

L1921 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinAbstractCoordinate

Documentation

The abstract endpoint after projection to one finite coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinAbstractCoordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(E / ℚ)

def

cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate

L1936 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate

Documentation

The relative infinite Artin symbol, restricted to a finite layer and then mapped to the corresponding rational coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(E / ℚ)

def

cyclotomicAbstractFixedFieldArtinFiniteCoordinate

L1949 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinFiniteCoordinate

Documentation

The finite relative Artin symbol mapped to one rational coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinFiniteCoordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(E / ℚ)

def

cyclotomicAbstractFixedFieldArtinRationalCoordinate

L1962 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinRationalCoordinate

Documentation

The finite rational Artin coordinate of the idele norm.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinRationalCoordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(E / ℚ)

theorem

cyclotomicAbstractFixedFieldArtinCoordinateNaturality

L1976 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateNaturality

Documentation

Naturality of the finite global Artin map at the concrete cyclotomic coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    cyclotomicAbstractFixedFieldArtinCoordinateNaturality
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E =
      cyclotomicAbstractFixedFieldArtinRationalCoordinate H a E

def

cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate

L1990 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate

Documentation

The rational endpoint after projection to one finite coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

def
    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Gal(E / ℚ)

theorem

cyclotomicAbstractFixedFieldArtinCoordinateRestriction

L2003 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRestriction

Documentation

The abstract restriction map projected to the concrete finite layer.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    cyclotomicAbstractFixedFieldArtinCoordinateRestriction
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
      cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E

theorem

cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection

L2034 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection

Documentation

Restricting the infinite relative Artin symbol supplies exactly the finite Artin coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E =
      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E

theorem

cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection

L2049 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection

Documentation

The rational cyclotomic Artin map projected to the same finite coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate H a E =
      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E

theorem

cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison

L2064 ↗

GlobalClassFieldTheory.Reciprocity.cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison

Documentation

Equality of the two full endpoints at one opaque finite coordinate.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a : IdeleGroup
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    AlgEquiv.restrictNormalHom E
        (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a) =
      AlgEquiv.restrictNormalHom E
        (cyclotomicAbstractFixedFieldArtinRationalEndpoint H a)

theorem

abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit

L2089 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit

Documentation

The finite-coordinate comparison assembled in the rational cyclotomic inverse limit.

Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
section FiniteCoordinateHelpers
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    FiniteDimensional ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  LocalClassFieldTheory.abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
  NumberField.of_module_finite ℚ
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (SeparableClosure ℚ) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomic_isAbelianGalois H
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ E :=
  E.toIntermediateField.algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField E :=
  NumberField.of_module_finite ℚ E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois ℚ E :=
  IsAbelianGalois.of_algHom E.toIntermediateField.val
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateNormal
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Normal ℚ E :=
  E.isGalois.to_normal
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  DivisionRing.toRatAlgebra
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
  let hI := rationalCyclotomicFieldInertia_le H.field
  apply IsScalarTower.of_algebraMap_eq
  intro x
  apply Subtype.ext
  change
    algebraMap ℚ
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI) x =
      algebraMap
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)
        (LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) hI)
        (algebraMap ℚ
          (LocalClassFieldTheory.abstractFixedField
            ℚ (SeparableClosure ℚ) H.field) x)
  exact
    IsScalarTower.algebraMap_apply
      ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)
      (LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI) x
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    NumberField
      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    SMul E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower ℚ E
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    Algebra
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  IntermediateField.toAlgebra
    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsScalarTower
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
noncomputable local instance
    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (E :
      FiniteGaloisIntermediateField
        ℚ rationalCyclotomicZHatField) :
    IsAbelianGalois
      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E

Lean source · statement

theorem
    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      IdeleGroup
        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
    cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a =
      cyclotomicAbstractFixedFieldArtinRationalEndpoint H a

theorem

abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom

L2113 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom

Documentation

The actual infinite global Artin map on an abstract fixed field restricts to the rational cyclotomic Artin map of the ordinary idele norm.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      IdeleGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let U :=
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI
    letI : IsAbelianGalois F U :=
      abstractFixedFieldCyclotomic_isAbelianGalois H
    abstractFixedFieldCyclotomicRestriction H
        (infiniteGlobalArtinMonoidHom F U a) =
      rationalCyclotomicZHatGlobalArtin
        (IdeleGroup.norm ℚ F a) := by
  exact
    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
      H a

theorem

abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom

L2146 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom

Documentation

In the normalized actual Galois coordinate, the infinite global Artin symbol is exactly the normalized cyclotomic idele value.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      IdeleGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let U :=
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI
    letI : IsAbelianGalois F U :=
      abstractFixedFieldCyclotomic_isAbelianGalois H
    Multiplicative.toAdd
        (abstractFixedFieldCyclotomicGalEquivZHat H
          (infiniteGlobalArtinMonoidHom F U a)) =
      normalizedCyclotomicZHatIdeleValue F
        (Additive.ofMul a) := by
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let : FiniteDimensional ℚ F :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  let : NumberField F :=
    NumberField.of_module_finite ℚ F
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) hI
  let : IsAbelianGalois F U :=
    abstractFixedFieldCyclotomic_isAbelianGalois H
  apply
    zHatMulNat_injective
      (H.residueDegree rationalCyclotomicDegreeData).pos
  calc
    (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
          Multiplicative.toAdd
            (abstractFixedFieldCyclotomicGalEquivZHat H
              (infiniteGlobalArtinMonoidHom F U a)) =
        Multiplicative.toAdd
          (rationalCyclotomicZHatFieldGalEquivZHat
            (abstractFixedFieldCyclotomicRestriction H
              (infiniteGlobalArtinMonoidHom F U a))) := by
      exact
        (abstractFixedFieldCyclotomicRestriction_coordinate H
          (infiniteGlobalArtinMonoidHom F U a)).symm
    _ =
        Multiplicative.toAdd
          (rationalCyclotomicZHatFieldGalEquivZHat
            (rationalCyclotomicZHatGlobalArtin
              (IdeleGroup.norm ℚ F a))) := by
      rw [
        abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom]
    _ =
        cyclotomicZHatNormComposite F
          (Additive.ofMul a) := by
      rfl
    _ =
        cyclotomicZHatIntersectionDegree F •
          normalizedCyclotomicZHatIdeleValue F
            (Additive.ofMul a) := by
      exact
        (cyclotomicZHatIntersectionDegree_nsmul_normalizedIdeleValue
          F (Additive.ofMul a)).symm
    _ =
        (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
          normalizedCyclotomicZHatIdeleValue F
            (Additive.ofMul a) := by
      rw [
        cyclotomicZHatIntersectionDegree_abstractFixedField_eq_residueDegree
          H]

theorem

abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk

L2233 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk

Documentation

Representative form of the actual cyclotomic Artin-coordinate identity after descent of the normalized value to the idele class group.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      IdeleGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let U :=
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI
    letI : IsAbelianGalois F U :=
      abstractFixedFieldCyclotomic_isAbelianGalois H
    Multiplicative.toAdd
        (abstractFixedFieldCyclotomicGalEquivZHat H
          (infiniteGlobalArtinMonoidHom F U a)) =
      normalizedCyclotomicZHatIdeleClassValueContinuous F
        (Additive.ofMul
          (QuotientGroup.mk'
            (IdeleGroup.principalSubgroup F) a)) := by
  have hSeparableClosureAlgebra :
      cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra =
        rationalSeparableClosureAlgebra :=
    Subsingleton.elim _ _
  cases hSeparableClosureAlgebra
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) hI
  let : IsAbelianGalois F U :=
    abstractFixedFieldCyclotomic_isAbelianGalois H
  calc
    Multiplicative.toAdd
        (abstractFixedFieldCyclotomicGalEquivZHat H
          (infiniteGlobalArtinMonoidHom F U a)) =
        normalizedCyclotomicZHatIdeleValue F
          (Additive.ofMul a) :=
      abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
        H a
    _ =
        normalizedCyclotomicZHatIdeleClassValueContinuous F
          (Additive.ofMul
            (QuotientGroup.mk'
              (IdeleGroup.principalSubgroup F) a)) :=
      (normalizedCyclotomicZHatIdeleClassValueContinuous_mk
        (K := F) a).symm

theorem

infiniteGlobalArtinMonoidHom_abstractFixedFieldCyclotomic_principalIdele

L2298 ↗

GlobalClassFieldTheory.Reciprocity.infiniteGlobalArtinMonoidHom_abstractFixedFieldCyclotomic_principalIdele

Documentation

The genuine infinite global Artin symbol of the cyclotomic maximal-unramified extension kills every principal idele of the abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    infiniteGlobalArtinMonoidHom_abstractFixedFieldCyclotomic_principalIdele
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (x :
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)ˣ) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let U :=
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI
    letI : IsAbelianGalois F U :=
      abstractFixedFieldCyclotomic_isAbelianGalois H
    infiniteGlobalArtinMonoidHom F U
        (IdeleGroup.principalIdele F x) =
      1 := by
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let : FiniteDimensional ℚ F :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  let : NumberField F :=
    NumberField.of_module_finite ℚ F
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) hI
  let : IsAbelianGalois F U :=
    abstractFixedFieldCyclotomic_isAbelianGalois H
  have hcoord :
      Multiplicative.toAdd
        (abstractFixedFieldCyclotomicGalEquivZHat H
          (infiniteGlobalArtinMonoidHom F U
            (IdeleGroup.principalIdele F x))) = 0 :=
    (abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
      H (IdeleGroup.principalIdele F x)).trans
      (normalizedCyclotomicZHatIdeleValue_principalIdele_eq_zero F x)
  apply (abstractFixedFieldCyclotomicGalEquivZHat H).injective
  rw [map_one]
  apply Multiplicative.ext
  exact hcoord.trans toAdd_one.symm

def

abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom

L2353 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom

Documentation

The genuine cyclotomic maximal-unramified Artin map descended to the idele class group of an abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · statement

def abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    IdeleClassGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field) →*
      Gal(
        LocalClassFieldTheory.abstractRelativeFixedField
          ℚ (SeparableClosure ℚ)
          (rationalCyclotomicFieldInertia_le H.field) /
        LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)

theorem

abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_mk

L2393 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_mk

Documentation

Evaluation of the descended maximal-unramified Artin map on an idele representative.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_mk
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      IdeleGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let U :=
      LocalClassFieldTheory.abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) hI
    letI : IsAbelianGalois F U :=
      abstractFixedFieldCyclotomic_isAbelianGalois H
    abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H
        (QuotientGroup.mk'
          (IdeleGroup.principalSubgroup F) a) =
      infiniteGlobalArtinMonoidHom F U a := by
  rw [abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom]
  exact QuotientGroup.lift_mk _ _ _

theorem

abstractFixedFieldCyclotomicGalEquivZHat_ideleClassArtinMonoidHom

L2425 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicGalEquivZHat_ideleClassArtinMonoidHom

Documentation

The descended genuine maximal-unramified Artin map is precisely the normalized cyclotomic idele-class value in the actual Galois coordinate.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicGalEquivZHat_ideleClassArtinMonoidHom
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (c :
      IdeleClassGroup
        (LocalClassFieldTheory.abstractFixedField
          ℚ (SeparableClosure ℚ) H.field)) :
    let F :=
      LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field
    letI : FiniteDimensional ℚ F :=
      LocalClassFieldTheory.abstractFixedField_finiteDimensional
        ℚ (SeparableClosure ℚ) H.field H.finite
    letI : NumberField F :=
      NumberField.of_module_finite ℚ F
    abstractFixedFieldCyclotomicGalEquivZHat H
        (abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H c) =
      normalizedCyclotomicZHatIdeleClassValueContinuousMul F c := by
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let : FiniteDimensional ℚ F :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  let : NumberField F :=
    NumberField.of_module_finite ℚ F
  have hSeparableClosureAlgebra :
      cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra =
        rationalSeparableClosureAlgebra :=
    Subsingleton.elim _ _
  cases hSeparableClosureAlgebra
  refine Quotient.inductionOn' c ?_
  intro a
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let U :=
    LocalClassFieldTheory.abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) hI
  let : IsAbelianGalois F U :=
    abstractFixedFieldCyclotomic_isAbelianGalois H
  calc
    abstractFixedFieldCyclotomicGalEquivZHat H
        (abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H
          (QuotientGroup.mk'
            (IdeleGroup.principalSubgroup F) a)) =
        abstractFixedFieldCyclotomicGalEquivZHat H
          (infiniteGlobalArtinMonoidHom F U a) :=
      congrArg (abstractFixedFieldCyclotomicGalEquivZHat H)
        (abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_mk H a)
    _ =
        normalizedCyclotomicZHatIdeleClassValueContinuousMul F
          (QuotientGroup.mk'
            (IdeleGroup.principalSubgroup F) a) := by
      apply Multiplicative.ext
      exact
        abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk
          H a

theorem

abstractFixedFieldCyclotomicIdeleClassArtin_eq_maximalUnramifiedNormResidue

L2489 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicIdeleClassArtin_eq_maximalUnramifiedNormResidue

Documentation

The genuine chosen-local-factor Artin map to the cyclotomic maximal-unramified extension is the abstract maximal-unramified norm-residue symbol. Both sides are characterized here by their common normalized valuation coordinate, so no finite reciprocity comparison is assumed.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicIdeleClassArtin_eq_maximalUnramifiedNormResidue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (a :
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation H.field) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let hnormal :
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI).Normal := by
      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
      infer_instance
    let qInertia :
        H.field.toSubgroup ⧸
            CyclicCohomology.extensionSubgroup H.field
              (rationalCyclotomicDegreeData.fieldInertia H.field)
              hI ≃*
          H.field.toSubgroup ⧸
            rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
      QuotientGroup.quotientMulEquivOfEq
        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
    abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H
        (Additive.toMul
          ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
              (hfinite := H.finite)).symm
            a)) =
      LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
        ℚ (SeparableClosure ℚ) H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI hnormal
        (qInertia.symm
          (ValuationData.maximalUnramifiedNormResidueSymbol
            rationalCyclotomicIdeleClassValuationData H a).toMul) := by
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let : FiniteDimensional ℚ F :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  let : NumberField F :=
    NumberField.of_module_finite ℚ F
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let qInertia :
      H.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI ≃*
        H.field.toSubgroup ⧸
          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
    QuotientGroup.quotientMulEquivOfEq
      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
  let c : IdeleClassGroup F :=
    Additive.toMul
      ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
          (hfinite := H.finite)).symm a)
  have hvaluation :
      ((rationalCyclotomicIdeleClassValuationData.valuationAt H a :
          rationalCyclotomicIdeleClassValuationData.valueGroup) : ZHat) =
        normalizedCyclotomicZHatIdeleClassValueContinuous F
          (Additive.ofMul c) := by
    simpa only [c, ofMul_toMul,
      (rationalAbstractFixedFieldIdeleClassEquivFixed H.field
        (hfinite := H.finite)).apply_symm_apply] using
      (rationalCyclotomicIdeleClassValuationData_valuationAt_fixed_apply
        H
        ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
          (hfinite := H.finite)).symm a))
  have hleft :
      Multiplicative.toAdd
        (abstractFixedFieldCyclotomicGalEquivZHat H
          (abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H c)) =
        normalizedCyclotomicZHatIdeleClassValueContinuous F
          (Additive.ofMul c) :=
    congrArg Multiplicative.toAdd
      (abstractFixedFieldCyclotomicGalEquivZHat_ideleClassArtinMonoidHom H c)
  have hright :
      ((rationalCyclotomicIdeleClassValuationData.valuationAt H a :
          rationalCyclotomicIdeleClassValuationData.valueGroup) : ZHat) =
        Multiplicative.toAdd
          (abstractFixedFieldCyclotomicGalEquivZHat H
            (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
              ℚ (SeparableClosure ℚ) H.field
              (rationalCyclotomicDegreeData.fieldInertia H.field)
              hI hnormal
              (qInertia.symm
                (ValuationData.maximalUnramifiedNormResidueSymbol
                  rationalCyclotomicIdeleClassValuationData H a).toMul))) := by
    rw [abstractFixedFieldCyclotomicGalEquivZHat_quotientClass]
    exact
      (ValuationData.maximalUnramifiedNormResidue_degree
        rationalCyclotomicIdeleClassValuationData H a).symm
  apply (abstractFixedFieldCyclotomicGalEquivZHat H).injective
  apply Multiplicative.ext
  exact hleft.trans (hvaluation.symm.trans hright)

theorem

abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_prime

L2596 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_prime

Documentation

A prime idele class has genuine maximal-unramified Artin symbol equal to the arithmetic Frobenius of its abstract fixed field.

Source context 10 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped IsMulCommutative NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open KummerTheory
/-- Keep this module on the rational algebra structures used by the
cyclotomic fixed-field API.  Generic intermediate-field instances are
propositionally equal here but not definitionally interchangeable. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
    Algebra ℚ (SeparableClosure ℚ) :=
  DivisionRing.toRatAlgebra
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
    Algebra ℚ rationalCyclotomicZHatField :=
  DivisionRing.toRatAlgebra
/-- On a quotient representative, cyclotomic restriction of the
actual relative automorphism is ordinary restriction of the same
ambient absolute-Galois automorphism. -/
noncomputable local instance
    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
    Normal ℚ rationalCyclotomicZHatField :=
  rationalCyclotomicZHatField_isNormal
/-- The fixed field attached to a finite abstract field is a number
field.  Keeping this as the single file-local instance makes it
available while later theorem binders are elaborated. -/
noncomputable local instance
    abstractFixedFieldCyclotomic_numberField
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
    NumberField
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) := by
  let : FiniteDimensional ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field) :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  exact
    NumberField.of_module_finite ℚ
      (LocalClassFieldTheory.abstractFixedField
        ℚ (SeparableClosure ℚ) H.field)

Lean source · full declaration

theorem
    abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom_prime
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (π :
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation H.field)
    (hπ :
      rationalCyclotomicIdeleClassValuationData.IsPrimeElement H π) :
    let hI :=
      rationalCyclotomicFieldInertia_le H.field
    let hnormal :
        (CyclicCohomology.extensionSubgroup H.field
          (rationalCyclotomicDegreeData.fieldInertia H.field)
          hI).Normal := by
      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
      infer_instance
    let qInertia :
        H.field.toSubgroup ⧸
            CyclicCohomology.extensionSubgroup H.field
              (rationalCyclotomicDegreeData.fieldInertia H.field)
              hI ≃*
          H.field.toSubgroup ⧸
            rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
      QuotientGroup.quotientMulEquivOfEq
        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
    abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H
        (Additive.toMul
          ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
              (hfinite := H.finite)).symm
            π)) =
      LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
        ℚ (SeparableClosure ℚ) H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI hnormal
        (qInertia.symm
          (rationalCyclotomicDegreeData.frobenius
            (H.toFiniteResidueAbstractField
              rationalCyclotomicDegreeData))) := by
  let F :=
    LocalClassFieldTheory.abstractFixedField
      ℚ (SeparableClosure ℚ) H.field
  let : FiniteDimensional ℚ F :=
    LocalClassFieldTheory.abstractFixedField_finiteDimensional
      ℚ (SeparableClosure ℚ) H.field H.finite
  let : NumberField F :=
    NumberField.of_module_finite ℚ F
  let hI :=
    rationalCyclotomicFieldInertia_le H.field
  let hnormal :
      (CyclicCohomology.extensionSubgroup H.field
        (rationalCyclotomicDegreeData.fieldInertia H.field)
        hI).Normal := by
    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
    infer_instance
  let qInertia :
      H.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI ≃*
        H.field.toSubgroup ⧸
          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
    QuotientGroup.quotientMulEquivOfEq
      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
  let c :
      IdeleClassGroup F :=
    Additive.toMul
      ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
          (hfinite := H.finite)).symm π)
  have hvalue :
      normalizedCyclotomicZHatIdeleClassValueContinuous F
          ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
              (hfinite := H.finite)).symm
            π) =
        1 := by
    calc
      normalizedCyclotomicZHatIdeleClassValueContinuous F
          ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
              (hfinite := H.finite)).symm
            π) =
          ((rationalCyclotomicIdeleClassValuationData.valuationAt H
              (rationalAbstractFixedFieldIdeleClassEquivFixed H.field
                (hfinite := H.finite)
                ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
                    (hfinite := H.finite)).symm
                  π)) :
              rationalCyclotomicIdeleClassValuationData.valueGroup) :
            ZHat) := by
        exact
          (rationalCyclotomicIdeleClassValuationData_valuationAt_fixed_apply
            H
            ((rationalAbstractFixedFieldIdeleClassEquivFixed H.field
                (hfinite := H.finite)).symm
              π)).symm
      _ =
          ((rationalCyclotomicIdeleClassValuationData.oneValue :
              rationalCyclotomicIdeleClassValuationData.valueGroup) :
            ZHat) := by
        rw [(rationalAbstractFixedFieldIdeleClassEquivFixed H.field
          (hfinite := H.finite)).apply_symm_apply]
        exact congrArg Subtype.val hπ
      _ = 1 :=
        rationalCyclotomicIdeleClassValuationData.oneValue_coe
  apply (abstractFixedFieldCyclotomicGalEquivZHat H).injective
  calc
    abstractFixedFieldCyclotomicGalEquivZHat H
        (abstractFixedFieldCyclotomicIdeleClassArtinMonoidHom H c) =
        normalizedCyclotomicZHatIdeleClassValueContinuousMul F c :=
      abstractFixedFieldCyclotomicGalEquivZHat_ideleClassArtinMonoidHom
        H c
    _ = Multiplicative.ofAdd (1 : ZHat) := by
      apply Multiplicative.ext
      exact hvalue
    _ =
        abstractFixedFieldCyclotomicGalEquivZHat H
          (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
            ℚ (SeparableClosure ℚ) H.field
            (rationalCyclotomicDegreeData.fieldInertia H.field)
            hI hnormal
            (qInertia.symm
              (rationalCyclotomicDegreeData.frobenius
                (H.toFiniteResidueAbstractField
                  rationalCyclotomicDegreeData)))) := by
      rw [
        abstractFixedFieldCyclotomicGalEquivZHat_quotientClass,
        rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv_frobenius]

Module documentation

Cyclotomic Artin coordinates over abstract fixed fields

Declarations