Module

AbstractFixedFieldGlobalNormResidue.lean

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidue

85 declarations indexed from sourceView source ↗

Module documentation

Global norm residue on actual fixed fields

An abstract finite abelian subextension of the rational absolute Galois group determines an actual finite abelian extension between its two fixed number fields. This file transports the abstract norm-residue symbol directly to the ordinary idele-class norm quotient of those fixed fields.

Keeping this construction in one ambient separable closure is essential for the norm--restriction diagrams: no independently chosen embedding of either field is introduced.

Imports2

Imported by3

def

rationalEmbeddingSeparableClosureAlgebra

L46 ↗

GlobalClassFieldTheory.Reciprocity.rationalEmbeddingSeparableClosureAlgebra

Documentation

The algebra structure on the rational separable closure induced by a specified rational field embedding. It is deliberately not an instance: different embeddings of the same field need not induce definitionally equal algebra structures.

Source context 11 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory

Lean source · statement

def rationalEmbeddingSeparableClosureAlgebra
    {F : Type} [Field F] [Algebra ℚ F]
    (i : F →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra F (SeparableClosure ℚ)

theorem

exists_numberFieldEmbeddingComparisonAutomorphism

L55 ↗

GlobalClassFieldTheory.Reciprocity.exists_numberFieldEmbeddingComparisonAutomorphism

Documentation

Two rational embeddings of the same number field into the fixed rational separable closure differ by an automorphism of that separable closure.

Source context 11 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory

Lean source · statement

theorem exists_numberFieldEmbeddingComparisonAutomorphism
    {F : Type} [Field F] [NumberField F]
    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
    ∃ σ : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ,
      ∀ x : F, σ (i x) = j x

def

numberFieldEmbeddingComparisonAutomorphism

L84 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddingComparisonAutomorphism

Documentation

The canonical comparison automorphism between two rational embeddings of one number field into the fixed separable closure.

Source context 11 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory

Lean source · statement

def numberFieldEmbeddingComparisonAutomorphism
    {F : Type} [Field F] [NumberField F]
    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ

theorem

numberFieldEmbeddingComparisonAutomorphism_apply

L94 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddingComparisonAutomorphism_apply

Documentation

The comparison automorphism carries the first embedded copy of the number field to the second one pointwise.

Source context 11 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory

Lean source · statement

theorem numberFieldEmbeddingComparisonAutomorphism_apply
    {F : Type} [Field F] [NumberField F]
    (i j : F →ₐ[ℚ] SeparableClosure ℚ)
    (x : F) :
    numberFieldEmbeddingComparisonAutomorphism i j (i x) =
      j x

theorem

conjugateClosedFixingSubgroup_embeddingRange

L106 ↗

GlobalClassFieldTheory.Reciprocity.conjugateClosedFixingSubgroup_embeddingRange

Documentation

Conjugating the fixing subgroup of one embedded copy of a number field by the comparison automorphism gives the fixing subgroup of the other embedded copy.

Source context 11 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory

Lean source · statement

theorem conjugateClosedFixingSubgroup_embeddingRange
    {F : Type} [Field F] [NumberField F]
    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
    conjugateClosedSubgroup
        (RamificationTheory.closedFixingSubgroup
          ℚ (SeparableClosure ℚ) i.fieldRange)
        (numberFieldEmbeddingComparisonAutomorphism j i) =
      RamificationTheory.closedFixingSubgroup
        ℚ (SeparableClosure ℚ) j.fieldRange

instance

numberFieldEmbeddedIdeleClassGroupIsMulCommutative

L153 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedIdeleClassGroupIsMulCommutative

Source context 12 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization

Lean source · statement

instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F)

instance

numberFieldEmbeddedIdeleClassSubgroupNormal

L158 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedIdeleClassSubgroupNormal

Source context 13 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩

Lean source · statement

instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal

def

numberFieldEmbeddedLowerEmbedding

L169 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedLowerEmbedding

Documentation

The lower embedding obtained by restricting an explicitly supplied embedding of the top field into the rational separable closure.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

def numberFieldEmbeddedLowerEmbedding
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    K →ₐ[ℚ] SeparableClosure ℚ

def

numberFieldEmbeddedSeparableClosureAlgebra

L179 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedSeparableClosureAlgebra

Documentation

The exact algebra structure on the rational separable closure induced by the lower embedding of an explicitly embedded number-field tower. Keeping this as a reducible definition lets every use of the associated separable- closure equivalence share one definitionally identical algebra structure.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

def numberFieldEmbeddedSeparableClosureAlgebra
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra K (SeparableClosure ℚ)

abbrev

numberFieldEmbeddedBaseSubgroup

L186 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseSubgroup

Documentation

The fixing subgroup of the explicitly embedded lower field.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

abbrev numberFieldEmbeddedBaseSubgroup
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)

abbrev

numberFieldEmbeddedTopSubgroup

L194 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedTopSubgroup

Documentation

The fixing subgroup of the explicitly embedded top field.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

abbrev numberFieldEmbeddedTopSubgroup
    (_K L : Type) [Field L] [NumberField L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)

theorem

numberFieldEmbeddedTopSubgroup_le_baseSubgroup

L202 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedTopSubgroup_le_baseSubgroup

Documentation

The top fixing subgroup lies in the lower fixing subgroup.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

theorem numberFieldEmbeddedTopSubgroup_le_baseSubgroup
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (numberFieldEmbeddedTopSubgroup K L j).toSubgroup ≤
      (numberFieldEmbeddedBaseSubgroup K L j).toSubgroup

def

numberFieldEmbeddedSeparableClosureEquiv

L218 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedSeparableClosureEquiv

Documentation

The separable closure of the actual lower field, identified with the rational separable closure carrying the algebra structure induced by an explicit compatible embedding.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · full declaration

def numberFieldEmbeddedSeparableClosureEquiv
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    letI : Algebra K (SeparableClosure ℚ) :=
      numberFieldEmbeddedSeparableClosureAlgebra K L j
    SeparableClosure K ≃ₐ[K] SeparableClosure ℚ := by
  letI : Algebra K (SeparableClosure ℚ) :=
    numberFieldEmbeddedSeparableClosureAlgebra K L j
  letI hScalarTower : IsScalarTower ℚ K (SeparableClosure ℚ) :=
    IsScalarTower.of_algebraMap_eq'
      (numberFieldEmbeddedLowerEmbedding K L j).comp_algebraMap.symm
  letI hseparable : Algebra.IsSeparable K (SeparableClosure ℚ) :=
    Algebra.isSeparable_tower_top_of_isSeparable
      ℚ K (SeparableClosure ℚ)
  letI hSepClosure : IsSepClosure K (SeparableClosure ℚ) :=
    ⟨IsSepClosure.sep_closed ℚ, hseparable⟩
  exact
    IsSepClosure.equiv K
      (SeparableClosure K) (SeparableClosure ℚ)

theorem

numberFieldEmbeddedExtensionSubgroup_normal

L239 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedExtensionSubgroup_normal

Documentation

The relative subgroup arising from an explicit compatible number-field embedding is normal.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

theorem numberFieldEmbeddedExtensionSubgroup_normal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal

instance

numberFieldEmbeddedExtensionSubgroupNormal

L260 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedExtensionSubgroupNormal

Documentation

The normality witness for an explicitly embedded tower, registered at the precise subgroup used by the downstream quotient constructions.

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]

Lean source · statement

instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal

theorem

numberFieldEmbeddedExtensionQuotient_finite

L272 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedExtensionQuotient_finite

Documentation

The relative quotient arising from an explicit compatible number-field embedding is finite.

Source context 16 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j

Lean source · statement

theorem numberFieldEmbeddedExtensionQuotient_finite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

instance

numberFieldEmbeddedExtensionQuotientFinite

L298 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedExtensionQuotientFinite

Documentation

The relative-index witness for an explicitly embedded tower, registered at the exact quotient consumed by FiniteNormQuotient.

Source context 16 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j

Lean source · statement

instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

abbrev

numberFieldEmbeddedFiniteAbstractField

L312 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteAbstractField

Documentation

The finite abstract field determined by the lower member of an explicitly embedded number-field tower.

Source context 17 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j

Lean source · full declaration

abbrev numberFieldEmbeddedFiniteAbstractField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) where
  field := numberFieldEmbeddedBaseSubgroup K L j
  finite := by
    simpa only [numberFieldEmbeddedBaseSubgroup] using
      (ambientEmbeddedAbsoluteQuotientFinite
        ℚ K (numberFieldEmbeddedLowerEmbedding K L j))

instance

numberFieldEmbeddedAbsoluteQuotientFinite

L324 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbsoluteQuotientFinite

Documentation

The absolute-index witness for the lower member of an explicitly embedded tower, registered at its specialized quotient type.

Source context 17 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j

Lean source · statement

instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j)))

abbrev

numberFieldEmbeddedFiniteGaloisSubextension

L340 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteGaloisSubextension

Documentation

The finite Galois subextension determined by an explicitly embedded number-field tower.

Source context 18 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite

Lean source · full declaration

abbrev numberFieldEmbeddedFiniteGaloisSubextension
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteGaloisSubextension
      (numberFieldEmbeddedBaseSubgroup K L j) where
  field := numberFieldEmbeddedTopSubgroup K L j
  below := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
  normal := numberFieldEmbeddedExtensionSubgroup_normal K L j
  finite := numberFieldEmbeddedExtensionQuotient_finite K L j

instance

numberFieldEmbeddedAbstractFixedFieldFiniteDimensional

L352 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractFixedFieldFiniteDimensional

Documentation

Shared finite-dimensional data for the fixed field of the lower subgroup in an explicitly embedded number-field tower.

Source context 18 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite

Lean source · statement

instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))

instance

numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional

L365 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional

Documentation

Shared relative finite-dimensional data for the two fixed fields of an explicitly embedded number-field tower.

Source context 19 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

instance

numberFieldEmbeddedAbstractFixedFieldScalarTower

L382 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractFixedFieldScalarTower

Source context 20 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)

Lean source · statement

instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

instance

numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional

L391 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional

Source context 21 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

instance

numberFieldEmbeddedAbstractFixedFieldNumberField

L404 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractFixedFieldNumberField

Source context 22 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

Lean source · statement

instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))

instance

numberFieldEmbeddedAbstractRelativeFixedFieldNumberField

L414 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldNumberField

Source context 23 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

instance

numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional

L425 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional

Source context 24 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ)

instance

numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField

L437 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ)

instance

numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra

L446 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra

Source context 26 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ)

instance

numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois

L459 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois

Source context 27 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra

Lean source · statement

instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))

def

numberFieldEmbeddedExtensionQuotientEquivGaloisGroup

L477 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedExtensionQuotientEquivGaloisGroup

Documentation

The quotient of the two explicitly embedded fixing subgroups is the actual relative Galois group.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def
    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient ≃*
      Gal(L / K)

def

numberFieldEmbeddedAbstractBaseFieldEquiv

L509 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractBaseFieldEquiv

Documentation

The original lower field is canonically equivalent to the fixed field of its explicitly embedded fixing subgroup.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedAbstractBaseFieldEquiv
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    K ≃ₐ[ℚ]
      abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)

def

numberFieldEmbeddedAbstractTopFieldEquiv

L521 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractTopFieldEquiv

Documentation

The original top field is canonically equivalent to the relative fixed field of its explicitly embedded fixing subgroup.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedAbstractTopFieldEquiv
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    L ≃ₐ[ℚ]
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup
          K L j)).restrictScalars ℚ

theorem

numberFieldEmbeddedAbstractFieldEquiv_algebraMap

L534 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbstractFieldEquiv_algebraMap

Documentation

The two explicit fixed-field equivalences commute with the tower algebra maps.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem numberFieldEmbeddedAbstractFieldEquiv_algebraMap
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (x : K) :
    numberFieldEmbeddedAbstractTopFieldEquiv K L j
        (algebraMap K L x) =
      algebraMap
        (abstractFixedField ℚ (SeparableClosure ℚ)
          (numberFieldEmbeddedBaseSubgroup K L j))
        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j x)

def

numberFieldEmbeddedIdeleClassEquivAmbientFixed

L551 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedIdeleClassEquivAmbientFixed

Documentation

The ordinary idele class group of the explicitly embedded lower field, transported to the fixed part of the rational absolute idele-class representation.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedIdeleClassEquivAmbientFixed
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Additive (IdeleClassGroup K) ≃+
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation
        (numberFieldEmbeddedBaseSubgroup K L j)

def

numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient

L566 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient

Documentation

The abstract finite norm quotient of an explicitly embedded tower is its genuine ordinary idele-class norm quotient.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def
    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteNormQuotient rationalIdeleClassRepresentation
        (numberFieldEmbeddedBaseSubgroup K L j)
        (numberFieldEmbeddedTopSubgroup K L j)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) ≃+
      Additive
        (IdeleClassGroup K ⧸
          (_root_.ideleClassNorm K L).range)

theorem

numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk

L595 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk
    {K₀ L₀ K₁ L₁ : Type}
    [Field K₀] [NumberField K₀]
    [Field L₀] [NumberField L₀] [Algebra K₀ L₀]
    [Field K₁] [NumberField K₁]
    [Field L₁] [NumberField L₁] [Algebra K₁ L₁]
    (eK : K₀ ≃ₐ[ℚ] K₁)
    (eL : L₀ ≃ₐ[ℚ] L₁)
    (h : ∀ x : K₀,
      eL (algebraMap K₀ L₀ x) =
        algebraMap K₁ L₁ (eK x))
    (c : IdeleClassGroup K₁) :
    (ordinaryIdeleClassNormQuotientCongrOfAlgEquiv eK eL h).symm
        (QuotientGroup.mk'
          (_root_.ideleClassNorm K₁ L₁).range c) =
      QuotientGroup.mk'
        (_root_.ideleClassNorm K₀ L₀).range
        ((ideleClassCongr eK).symm c)

def

numberFieldEmbeddedFiniteNormClassPublicValue

L619 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormClassPublicValue

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedFiniteNormClassPublicValue
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    Additive
      (IdeleClassGroup K ⧸
        (_root_.ideleClassNorm K L).range)

def

numberFieldEmbeddedFiniteNormClassExpectedValue

L635 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormClassExpectedValue

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedFiniteNormClassExpectedValue
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    Additive
      (IdeleClassGroup K ⧸
        (_root_.ideleClassNorm K L).range)

def

numberFieldEmbeddedFiniteNormClassDirectComparisonValue

L649 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormClassDirectComparisonValue

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def
    numberFieldEmbeddedFiniteNormClassDirectComparisonValue
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    Additive
      (IdeleClassGroup K ⧸
        (_root_.ideleClassNorm K L).range)

theorem

numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison

L678 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem
    numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    numberFieldEmbeddedFiniteNormClassPublicValue K L j a =
      numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a

def

numberFieldEmbeddedActualNormClassRepresentativeValue

L691 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedActualNormClassRepresentativeValue

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

def numberFieldEmbeddedActualNormClassRepresentativeValue
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    Additive
      (IdeleClassGroup K ⧸
        (_root_.ideleClassNorm K L).range)

theorem

numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue

L716 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem
    numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a =
      numberFieldEmbeddedActualNormClassRepresentativeValue K L j a

theorem

numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected

L740 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    numberFieldEmbeddedActualNormClassRepresentativeValue K L j a =
      numberFieldEmbeddedFiniteNormClassExpectedValue K L j a

theorem

numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass

L781 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass

Documentation

On a finite norm-class representative, the explicit fixed-field comparison is the genuine ordinary idele-class quotient.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem
    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)) :
    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
        (finiteNormClass rationalIdeleClassRepresentation
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) a) =
      Additive.ofMul
        (QuotientGroup.mk'
          (_root_.ideleClassNorm K L).range
          (Additive.toMul
            ((numberFieldEmbeddedIdeleClassEquivAmbientFixed K L j).symm a)))

theorem

numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass

L812 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass

Documentation

On an ordinary idele class, the explicit fixed-part realization followed by the abstract finite norm-class map is the genuine quotient class modulo the ordinary idele-class norm.

Source context 28 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)

Lean source · statement

theorem
    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (c : IdeleClassGroup K) :
    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
        (finiteNormClass rationalIdeleClassRepresentation
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
          (numberFieldEmbeddedIdeleClassEquivAmbientFixed
            K L j (Additive.ofMul c))) =
      Additive.ofMul
        (QuotientGroup.mk'
          (_root_.ideleClassNorm K L).range c)

def

numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup

L837 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup

Documentation

The abelianized quotient of the explicitly embedded tower is the actual abelian Galois group.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

def
    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Additive
      (Abelianization
        (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient) ≃+
      Additive Gal(L / K)

def

globalNormResidueEquivOfEmbedding

L854 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueEquivOfEmbedding

Documentation

The actual global norm-residue equivalence constructed from an explicit compatible embedding of a finite abelian number-field extension into the rational separable closure.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

def globalNormResidueEquivOfEmbedding
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Additive
        (IdeleClassGroup K ⧸
          (_root_.ideleClassNorm K L).range) ≃+
      Additive Gal(L / K)

theorem

globalNormResidueEquivOfEmbedding_finiteNormClass

L882 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueEquivOfEmbedding_finiteNormClass

Documentation

The explicit-embedding norm-residue equivalence on a finite norm class.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

theorem globalNormResidueEquivOfEmbedding_finiteNormClass
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (x : FiniteNormQuotient rationalIdeleClassRepresentation
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :
    globalNormResidueEquivOfEmbedding K L j
        (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
          K L j x) =
      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup K L j
        (rationalCyclotomicDegreeData.normResidueSymbol
          rationalIdeleClassRepresentation
          rationalCyclotomicIdeleClassValuationData
          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
          (numberFieldEmbeddedFiniteAbstractField K L j)
          (numberFieldEmbeddedFiniteGaloisSubextension K L j) x)

def

globalNormResidueMonoidHomOfEmbedding

L903 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueMonoidHomOfEmbedding

Documentation

The global norm-residue homomorphism obtained from an explicit compatible embedding.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

def globalNormResidueMonoidHomOfEmbedding
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IdeleClassGroup K →* Gal(L / K)

theorem

globalNormResidueMonoidHomOfEmbedding_apply

L921 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueMonoidHomOfEmbedding_apply

Documentation

Evaluation of the explicit-embedding global norm-residue map is the abstract norm-residue symbol evaluated on the corresponding genuine fixed-part finite norm class.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

theorem globalNormResidueMonoidHomOfEmbedding_apply
    (j : L →ₐ[ℚ] SeparableClosure ℚ)
    (c : IdeleClassGroup K) :
    globalNormResidueMonoidHomOfEmbedding K L j c =
      Additive.toMul
        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
          K L j
          (rationalCyclotomicDegreeData.normResidueSymbol
            rationalIdeleClassRepresentation
            rationalCyclotomicIdeleClassValuationData
            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
            (numberFieldEmbeddedFiniteAbstractField K L j)
            (numberFieldEmbeddedFiniteGaloisSubextension K L j)
            (finiteNormClass rationalIdeleClassRepresentation
              (numberFieldEmbeddedBaseSubgroup K L j)
              (numberFieldEmbeddedTopSubgroup K L j)
              (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
              (numberFieldEmbeddedIdeleClassEquivAmbientFixed
                K L j (Additive.ofMul c)))))

theorem

numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard

L968 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard

Documentation

The ambient-fixed idèle-class transport for the standard embedding is the same map as the transport for an explicitly supplied embedding. This comparison is kept at the transport boundary, before forming norm quotients or applying reciprocity.

Source context 30 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]
omit [FiniteDimensional K L] [IsAbelianGalois K L] in

Lean source · statement

theorem numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard :
    numberFieldTowerIdeleClassEquivAmbientFixed K L =
      numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
        (numberFieldSeparableClosureEmbedding L)

theorem

numberFieldTowerNormResidueValue_eq_embedded_standard

L987 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldTowerNormResidueValue_eq_embedded_standard

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

theorem numberFieldTowerNormResidueValue_eq_embedded_standard
    (c : IdeleClassGroup K) :
    numberFieldTowerAbelianizedExtensionQuotientEquivGaloisGroup K L
        (rationalCyclotomicDegreeData.normResidueSymbol
          rationalIdeleClassRepresentation
          rationalCyclotomicIdeleClassValuationData
          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
          (numberFieldTowerReciprocityFiniteAbstractField K L)
          (numberFieldTowerFiniteGaloisSubextension K L)
          (finiteNormClass rationalIdeleClassRepresentation
            (numberFieldTowerBaseSubgroup K L)
            (numberFieldTowerTopSubgroup L)
            (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
            (numberFieldTowerIdeleClassEquivAmbientFixed K L (Additive.ofMul c)))) =
      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
        K L (numberFieldSeparableClosureEmbedding L)
        (rationalCyclotomicDegreeData.normResidueSymbol
          rationalIdeleClassRepresentation
          rationalCyclotomicIdeleClassValuationData
          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
          (numberFieldEmbeddedFiniteAbstractField K L
            (numberFieldSeparableClosureEmbedding L))
          (numberFieldEmbeddedFiniteGaloisSubextension K L
            (numberFieldSeparableClosureEmbedding L))
          (finiteNormClass rationalIdeleClassRepresentation
            (numberFieldEmbeddedBaseSubgroup K L
              (numberFieldSeparableClosureEmbedding L))
            (numberFieldEmbeddedTopSubgroup K L
              (numberFieldSeparableClosureEmbedding L))
            (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L
              (numberFieldSeparableClosureEmbedding L))
            (numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
              (numberFieldSeparableClosureEmbedding L) (Additive.ofMul c))))

theorem

globalNormResidueEquiv_eq_ofEmbedding_standard

L1050 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueEquiv_eq_ofEmbedding_standard

Documentation

At the standard embedding, the two global norm-residue equivalences agree on the actual norm quotient. The comparison is extensional: it uses surjectivity of the quotient map and the established evaluation formulas, not definitional equality of the two quotient constructions.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

theorem globalNormResidueEquiv_eq_ofEmbedding_standard :
    globalNormResidueEquiv K L =
      globalNormResidueEquivOfEmbedding K L
        (numberFieldSeparableClosureEmbedding L)

theorem

globalNormResidueMonoidHom_eq_ofEmbedding_standard

L1078 ↗

GlobalClassFieldTheory.Reciprocity.globalNormResidueMonoidHom_eq_ofEmbedding_standard

Documentation

The existing global norm-residue map is the explicit-embedding construction for the standard chosen embedding of the top field.

Source context 29 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRealization
local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance numberFieldEmbeddedIdeleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    (K L : Type) [Field K] [NumberField K]
    [Field L] [NumberField L] [Algebra K L]
/-- The normality witness for an explicitly embedded tower, registered at
the precise subgroup used by the downstream quotient constructions. -/
noncomputable local instance
    numberFieldEmbeddedExtensionSubgroupNormal
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup K L j)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
  numberFieldEmbeddedExtensionSubgroup_normal K L j
/-- The relative-index witness for an explicitly embedded tower, registered
at the exact quotient consumed by `FiniteNormQuotient`. -/
noncomputable local instance
    numberFieldEmbeddedExtensionQuotientFinite
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup K L j)
          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  numberFieldEmbeddedExtensionQuotient_finite K L j
/-- The absolute-index witness for the lower member of an explicitly embedded
tower, registered at its specialized quotient type. -/
noncomputable local instance
    numberFieldEmbeddedAbsoluteQuotientFinite
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          (numberFieldEmbeddedBaseSubgroup K L j)
          (le_baseField
            (numberFieldEmbeddedBaseSubgroup K L j))) :=
  (numberFieldEmbeddedFiniteAbstractField K L j).finite
/-- Shared finite-dimensional data for the fixed field of the lower subgroup
in an explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
/-- Shared relative finite-dimensional data for the two fixed fields of an
explicitly embedded number-field tower. -/
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
    (numberFieldEmbeddedExtensionQuotientFinite K L j)
local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractFixedFieldNumberField
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
  change FiniteDimensional ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
  infer_instance
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  NumberField.of_module_finite ℚ _
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    Algebra
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
  (IntermediateField.inclusion
    (abstractFixedField_le ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
noncomputable local instance
    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
    [FiniteDimensional K L] [IsGalois K L]
    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L j))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup K L j)
    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
variable [FiniteDimensional K L] [IsAbelianGalois K L]

Lean source · statement

theorem globalNormResidueMonoidHom_eq_ofEmbedding_standard :
    globalNormResidueMonoidHom K L =
      globalNormResidueMonoidHomOfEmbedding K L
        (numberFieldSeparableClosureEmbedding L)

theorem

numberFieldEmbeddedBaseSubgroup_le_of_tower

L1119 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseSubgroup_le_of_tower

Documentation

A compatible common embedding reverses the inclusion of the two base fields into an inclusion of their fixing subgroups.

Source context 13 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRestriction
variable
    (K K' L L' : Type)
    [Field K] [NumberField K]
    [Field K'] [NumberField K']
    [Field L] [NumberField L]
    [Field L'] [NumberField L']
    [Algebra K K'] [Algebra K L] [Algebra K L']
    [Algebra K' L'] [Algebra L L']
    [IsScalarTower K K' L'] [IsScalarTower K L L']

Lean source · full declaration

theorem numberFieldEmbeddedBaseSubgroup_le_of_tower
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
      j.comp (IsScalarTower.toAlgHom ℚ L L')
    (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup ≤
      (numberFieldEmbeddedBaseSubgroup K L jLower).toSubgroup := by
  dsimp only
  change
    (numberFieldEmbeddedLowerEmbedding K' L' j).fieldRange.fixingSubgroup ≤
      (numberFieldEmbeddedLowerEmbedding K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup
  apply
    (numberFieldEmbeddedLowerEmbedding K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup_le
  intro x hx
  rcases hx with ⟨y, rfl⟩
  refine ⟨algebraMap K K' y, ?_⟩
  change
    j (algebraMap K' L' (algebraMap K K' y)) =
      j (algebraMap L L' (algebraMap K L y))
  rw [← IsScalarTower.algebraMap_apply K K' L',
    ← IsScalarTower.algebraMap_apply K L L']

theorem

numberFieldEmbeddedTopSubgroup_le_of_tower

L1148 ↗

GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedTopSubgroup_le_of_tower

Documentation

A compatible common embedding reverses the inclusion of the two top fields into an inclusion of their fixing subgroups.

Source context 14 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
section EmbeddedNumberFieldRestriction
variable
    (K K' L L' : Type)
    [Field K] [NumberField K]
    [Field K'] [NumberField K']
    [Field L] [NumberField L]
    [Field L'] [NumberField L']
    [Algebra K K'] [Algebra K L] [Algebra K L']
    [Algebra K' L'] [Algebra L L']
    [IsScalarTower K K' L'] [IsScalarTower K L L']
omit [Field K] [NumberField K]
    [Field K'] [NumberField K']
    [Algebra K K'] [Algebra K L] [Algebra K L'] [Algebra K' L']
    [IsScalarTower K K' L'] [IsScalarTower K L L'] in

Lean source · full declaration

theorem numberFieldEmbeddedTopSubgroup_le_of_tower
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
      j.comp (IsScalarTower.toAlgHom ℚ L L')
    (numberFieldEmbeddedTopSubgroup K' L' j).toSubgroup ≤
      (numberFieldEmbeddedTopSubgroup K L jLower).toSubgroup := by
  dsimp only
  change
    j.fieldRange.fixingSubgroup ≤
      (j.comp
        (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup
  apply
    (j.comp
      (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup_le
  intro x hx
  rcases hx with ⟨y, rfl⟩
  exact ⟨algebraMap L L' y, rfl⟩

instance

abstractFixedFieldBaseQuotientFinite

L1173 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldBaseQuotientFinite

Source context 12 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)

Lean source · statement

instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field))

instance

abstractFixedFieldRelativeQuotientFinite

L1183 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldRelativeQuotientFinite

Source context 13 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite

Lean source · statement

instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below)

instance

abstractFixedFieldRelativeQuotientIsMulCommutative

L1189 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldRelativeQuotientIsMulCommutative

Source context 14 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite

Lean source · statement

instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient

instance

abstractFixedFieldFiniteDimensional

L1193 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldFiniteDimensional

Source context 15 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative

Lean source · statement

instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)

instance

abstractRelativeFixedFieldFiniteDimensional

L1199 ↗

GlobalClassFieldTheory.Reciprocity.abstractRelativeFixedFieldFiniteDimensional

Source context 16 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite

Lean source · statement

instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractFixedFieldRelativeScalarTower

L1208 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldRelativeScalarTower

Source context 17 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite

Lean source · statement

instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractRelativeFixedFieldAbsoluteFiniteDimensional

L1215 ↗

GlobalClassFieldTheory.Reciprocity.abstractRelativeFixedFieldAbsoluteFiniteDimensional

Source context 18 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractFixedFieldNumberField

L1224 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldNumberField

Source context 19 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)

Lean source · statement

instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)

instance

abstractFixedFieldIdeleClassGroupIsMulCommutative

L1233 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldIdeleClassGroupIsMulCommutative

Documentation

The lower fixed idèle-class group is commutative. Naming the mixin before the public quotient declarations avoids delayed normality synthesis inside their definition bodies.

Source context 20 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)

Lean source · statement

instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field))

instance

abstractRelativeFixedFieldNumberField

L1240 ↗

GlobalClassFieldTheory.Reciprocity.abstractRelativeFixedFieldNumberField

Source context 21 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩

Lean source · statement

instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractRelativeFixedFieldIsGalois

L1251 ↗

GlobalClassFieldTheory.Reciprocity.abstractRelativeFixedFieldIsGalois

Documentation

Use the same explicit Galois witness as the fixed-field quotient comparison. Deriving it through IsAbelianGalois produces an equivalent but much larger dependent instance path.

Source context 22 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)

Lean source · statement

instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractRelativeFixedFieldIsAbelianGalois

L1261 ↗

GlobalClassFieldTheory.Reciprocity.abstractRelativeFixedFieldIsAbelianGalois

Source context 23 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal

Lean source · statement

instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)

instance

abstractFixedFieldIdeleClassNormRangeNormal

L1270 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldIdeleClassNormRangeNormal

Documentation

Use one opaque normality witness for the actual fixed-field norm range. This keeps every occurrence of its quotient group on the same instance path.

Source context 24 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L

Lean source · statement

instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal

def

abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup

L1280 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup

Documentation

The abelianized abstract extension quotient is the actual Galois group of the corresponding pair of fixed fields.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

def
    abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below
    Additive
        (Abelianization
          (FiniteGaloisSubextension.extensionQuotient
            L.toFiniteGaloisExtension)) ≃+
      Additive (Gal(E / F)) := by
  dsimp only
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) K.field
  let E :=
    abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below
  let e :
      L.extensionQuotient ≃*
        Gal(E / F) :=
    L.extensionQuotientMulEquiv.trans
      (abstractExtensionQuotientEquivGaloisGroup
        ℚ (SeparableClosure ℚ)
        K.field L.field L.below L.normal)
  exact
    MulEquiv.toAdditive
      ((Abelianization.equivOfComm :
          L.extensionQuotient ≃*
            Abelianization L.extensionQuotient).symm.trans e)

def

abstractFixedFieldGlobalNormResidueEquiv

L1317 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueEquiv

Documentation

The actual fixed-field global norm-residue equivalence

C_F / N_{E/F} C_E ≃ Gal(E/F)

attached to an abstract finite abelian subextension in the rational absolute class formation.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

def abstractFixedFieldGlobalNormResidueEquiv :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below
    Additive
        (IdeleClassGroup F ⧸
          (_root_.ideleClassNorm F E).range) ≃+
      Additive (Gal(E / F)) := by
  dsimp only
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) K.field
  let E :=
    abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below
  let eNorm :
      FiniteNormQuotient rationalIdeleClassRepresentation
          K.field L.field L.below ≃+
        Additive
          (Abelianization L.toFiniteGaloisExtension.extensionQuotient) :=
    rationalCyclotomicDegreeData.normResidueSymbol
      rationalIdeleClassRepresentation
      rationalCyclotomicIdeleClassValuationData
      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
      K L.toFiniteGaloisExtension
  exact
    (rationalFiniteNormQuotientEquivIdeleClassNormQuotient
        K.field L.field L.below L.normal).symm.trans
      (eNorm.trans
        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
          K L))

def

abstractFixedFieldFiniteNormResidueGaloisEquiv

L1352 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldFiniteNormResidueGaloisEquiv

Documentation

The abstract finite norm-residue equivalence with its dependent source instance fixed to the public finite norm quotient.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

def abstractFixedFieldFiniteNormResidueGaloisEquiv :
    FiniteNormQuotient rationalIdeleClassRepresentation
        K.field L.field L.below ≃+
      Additive
        (Gal(
          (abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) L.below) /
          (abstractFixedField ℚ (SeparableClosure ℚ) K.field)))

def

ambientFixedGlobalNormResidueAddMonoidHom

L1390 ↗

GlobalClassFieldTheory.Reciprocity.ambientFixedGlobalNormResidueAddMonoidHom

Documentation

The abstract norm-residue symbol on the fixed part of the rational absolute idele-class representation, with its value transported to the actual Galois group of the two fixed fields. This is the form consumed directly by the abstract norm--restriction naturality theorem.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

def ambientFixedGlobalNormResidueAddMonoidHom :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below
    ambientFixedAddSubgroup
        rationalIdeleClassRepresentation K.field →+
      Additive (Gal(E / F)) := by
  dsimp only
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) K.field
  let E :=
    abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below
  exact
    (abstractFixedFieldFiniteNormResidueGaloisEquiv K L).toAddMonoidHom.comp
      (finiteNormClassHom rationalIdeleClassRepresentation
        K.field L.field L.below)

def

abstractFixedFieldIdeleClassToAmbientFixedMonoidHom

L1412 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldIdeleClassToAmbientFixedMonoidHom

Documentation

The ordinary idele class group of the lower fixed field, transported to the fixed part of the rational absolute idele-class representation.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

def abstractFixedFieldIdeleClassToAmbientFixedMonoidHom :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    IdeleClassGroup F →*
      Multiplicative
        (ambientFixedAddSubgroup
          rationalIdeleClassRepresentation K.field) :=
  (rationalAbstractFixedFieldIdeleClassEquivFixed
    K.field).toAddMonoidHom.toMultiplicativeRight

def

abstractFixedFieldGlobalNormResidueMonoidHom

L1425 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom

Documentation

The actual norm-residue homomorphism on the ordinary idele class group of the lower fixed field, constructed without choosing a second field embedding.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

def abstractFixedFieldGlobalNormResidueMonoidHom :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below
    IdeleClassGroup F →* Gal(E / F) :=
  (ambientFixedGlobalNormResidueAddMonoidHom K L).toMultiplicative.comp
    (abstractFixedFieldIdeleClassToAmbientFixedMonoidHom K)

theorem

ambientFixedGlobalNormResidueAddMonoidHom_apply

L1436 ↗

GlobalClassFieldTheory.Reciprocity.ambientFixedGlobalNormResidueAddMonoidHom_apply

Documentation

Pointwise form of the ambient fixed-part norm-residue homomorphism.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem ambientFixedGlobalNormResidueAddMonoidHom_apply
    (a :
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation K.field) :
    ambientFixedGlobalNormResidueAddMonoidHom K L a =
      abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
        K L
        (rationalCyclotomicDegreeData.normResidueSymbol
          rationalIdeleClassRepresentation
          rationalCyclotomicIdeleClassValuationData
          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
          K L.toFiniteGaloisExtension
          (finiteNormClass rationalIdeleClassRepresentation
            K.field L.field L.below a))

theorem

abstractFixedFieldGlobalNormResidueMonoidHom_apply

L1457 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom_apply

Documentation

Pointwise form of the transported fixed-field norm-residue homomorphism.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · full declaration

theorem abstractFixedFieldGlobalNormResidueMonoidHom_apply :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) K.field
    ∀ c : IdeleClassGroup F,
      abstractFixedFieldGlobalNormResidueMonoidHom K L c =
        Additive.toMul
          (ambientFixedGlobalNormResidueAddMonoidHom K L
            ((rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
              (Additive.ofMul c))) := by
  dsimp only
  intro c
  rfl

theorem

abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply

L1474 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply

Documentation

Transporting an ordinary fixed-field idele class to the ambient fixed part and applying the abstract norm-residue map gives exactly the actual fixed-field norm-residue value.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply
    (a :
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation K.field) :
    abstractFixedFieldGlobalNormResidueMonoidHom K L
        (Additive.toMul
          ((rationalAbstractFixedFieldIdeleClassEquivFixed
            K.field).symm a)) =
      Additive.toMul
        (ambientFixedGlobalNormResidueAddMonoidHom K L a)

theorem

ambientFixedGlobalNormResidueAddMonoidHom_eq_zero_iff

L1498 ↗

GlobalClassFieldTheory.Reciprocity.ambientFixedGlobalNormResidueAddMonoidHom_eq_zero_iff

Documentation

The ambient fixed-part reciprocity value vanishes precisely when its finite norm class vanishes.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem ambientFixedGlobalNormResidueAddMonoidHom_eq_zero_iff
    (a :
      ambientFixedAddSubgroup
        rationalIdeleClassRepresentation K.field) :
    ambientFixedGlobalNormResidueAddMonoidHom K L a = 0 ↔
      finiteNormClass rationalIdeleClassRepresentation
        K.field L.field L.below a = 0

theorem

rationalAbstractFixedFieldIdeleClassEquivFixed_mem_finiteNormSubgroup_iff

L1517 ↗

GlobalClassFieldTheory.Reciprocity.rationalAbstractFixedFieldIdeleClassEquivFixed_mem_finiteNormSubgroup_iff

Documentation

The fixed-field idele-class comparison carries the abstract finite norm subgroup exactly to the ordinary norm range.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem
    rationalAbstractFixedFieldIdeleClassEquivFixed_mem_finiteNormSubgroup_iff
    (c : IdeleClassGroup
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :
    (rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
          (Additive.ofMul c) ∈
        finiteNormSubgroup rationalIdeleClassRepresentation
          K.field L.field L.below ↔
      c ∈
        (_root_.ideleClassNorm
          (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
          (abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) L.below)).range

theorem

abstractFixedFieldGlobalNormResidueMonoidHom_eq_one_iff

L1568 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom_eq_one_iff

Documentation

Triviality of the fixed-field norm-residue symbol is exactly membership in the actual ordinary idele-class norm range.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem abstractFixedFieldGlobalNormResidueMonoidHom_eq_one_iff
    (c : IdeleClassGroup
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :
      abstractFixedFieldGlobalNormResidueMonoidHom K L c = 1 ↔
        c ∈
          (_root_.ideleClassNorm
            (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
            (abstractRelativeFixedField
              ℚ (SeparableClosure ℚ) L.below)).range

theorem

ambientFixedGlobalNormResidueAddMonoidHom_surjective

L1619 ↗

GlobalClassFieldTheory.Reciprocity.ambientFixedGlobalNormResidueAddMonoidHom_surjective

Documentation

The ambient fixed-part reciprocity homomorphism is surjective.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem ambientFixedGlobalNormResidueAddMonoidHom_surjective :
    Function.Surjective
      (ambientFixedGlobalNormResidueAddMonoidHom K L)

theorem

abstractFixedFieldGlobalNormResidueMonoidHom_surjective

L1634 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom_surjective

Documentation

The fixed-field global norm-residue homomorphism is surjective onto the actual Galois group.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem abstractFixedFieldGlobalNormResidueMonoidHom_surjective :
    Function.Surjective
      (abstractFixedFieldGlobalNormResidueMonoidHom K L)

theorem

abstractFixedFieldGlobalNormResidueMonoidHom_ker

L1650 ↗

GlobalClassFieldTheory.Reciprocity.abstractFixedFieldGlobalNormResidueMonoidHom_ker

Documentation

The kernel of the fixed-field global norm-residue homomorphism is the genuine ordinary idele-class norm range.

Source context 25 commands

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

open scoped IsMulCommutative NumberField
open NumberField
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
variable
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteAbelianSubextension K.field)
local instance abstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          K.field (le_baseField K.field)) :=
  K.finite
local instance abstractFixedFieldRelativeQuotientFinite :
    Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
  L.finite
local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
    IsMulCommutative L.extensionQuotient :=
  L.commutative
noncomputable local instance abstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K.field K.finite
noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below K.finite L.finite
local instance abstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
noncomputable local instance abstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
/-- The lower fixed idèle-class group is commutative.  Naming the mixin
before the public quotient declarations avoids delayed normality synthesis
inside their definition bodies. -/
local instance
    abstractFixedFieldIdeleClassGroupIsMulCommutative :
    IsMulCommutative
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
noncomputable local instance abstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) L.below)
/-- Use the same explicit Galois witness as the fixed-field quotient
comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
but much larger dependent instance path. -/
noncomputable local instance
    abstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    K.field L.field L.below L.normal
noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
/-- Use one opaque normality witness for the actual fixed-field norm range.
This keeps every occurrence of its quotient group on the same instance path. -/
local instance
    abstractFixedFieldIdeleClassNormRangeNormal :
    ((_root_.ideleClassNorm
      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
  infer_instance

Lean source · statement

theorem abstractFixedFieldGlobalNormResidueMonoidHom_ker :
    (abstractFixedFieldGlobalNormResidueMonoidHom K L).ker =
      (_root_.ideleClassNorm
        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) L.below)).range

Module documentation

Global norm residue on actual fixed fields

Declarations