NYLL
Module

GlobalNormResidueNaturality.lean

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturality
96 declarations indexed from sourceView source ↗

Module documentation

Naturality of the global norm-residue symbol

This file records the same-base restriction specialization of abstract norm-residue naturality in the rational absolute class formation. All closed subgroups remain in one fixed separable-closure ambient, so the statement is directly usable by fixed-field overextension arguments.

Imports2
Imported by5

Declarations

def

naturalityIdeleClassCommGroup

L38 ↗
GlobalClassFieldTheory.Reciprocity.naturalityIdeleClassCommGroup
Source context 11 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
Lean source · statement
def naturalityIdeleClassCommGroup
    (F : Type) [Field F] [NumberField F] : CommGroup (IdeleClassGroup F)
instance

ideleClassGroupIsMulCommutative

L44 ↗
GlobalClassFieldTheory.Reciprocity.ideleClassGroupIsMulCommutative
Source context 12 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
Lean source · statement
instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F)
instance

ideleClassSubgroupNormal

L49 ↗
GlobalClassFieldTheory.Reciprocity.ideleClassSubgroupNormal
Source context 13 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
Lean source · statement
instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal
theorem

finiteGaloisSubextension_eq_of_field_eq

L57 ↗
GlobalClassFieldTheory.Reciprocity.finiteGaloisSubextension_eq_of_field_eq
Documentation

Two finite Galois subextensions with the same underlying closed subgroup are equal; the remaining structure fields are proof-irrelevant.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem finiteGaloisSubextension_eq_of_field_eq
    {G : Type u} [Group G] [TopologicalSpace G]
    {K : ClosedSubgroup G}
    (A B : FiniteGaloisSubextension K)
    (h : A.field = B.field) :
    A = B
theorem

finiteGaloisSubextension_transport_eq_of_field_eq

L74 ↗
GlobalClassFieldTheory.Reciprocity.finiteGaloisSubextension_transport_eq_of_field_eq
Documentation

Rebase a finite Galois subextension along equality of its bundled base. The field equality is the only data component; the remaining fields are proof-irrelevant.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
theorem finiteGaloisSubextension_transport_eq_of_field_eq
    {G : Type u} [Group G] [TopologicalSpace G]
    {A B : FiniteAbstractField G}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    (Q : FiniteGaloisSubextension B.field)
    (hfield : P.field = Q.field) :
    Eq.mp
      (congrArg
        (fun X : FiniteAbstractField G =>
          FiniteGaloisSubextension X.field)
        hAB)
      P = Q := by
  cases hAB
  exact finiteGaloisSubextension_eq_of_field_eq P Q hfield
theorem

rationalAmbientFixedAddEquiv_transport_apply_val

L92 ↗
GlobalClassFieldTheory.Reciprocity.rationalAmbientFixedAddEquiv_transport_apply_val
Documentation

Transporting an additive equivalence between rational ambient fixed subgroups does not change the underlying direct-limit class.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
theorem rationalAmbientFixedAddEquiv_transport_apply_val
    {A B : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
    (hAB : A = B)
    {X : Type} [AddGroup X]
    (e : X ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
    (x : X) :
    ((Eq.mp
        (congrArg
          (fun Y : FiniteAbstractField
              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
            X ≃+ ambientFixedAddSubgroup
              rationalIdeleClassRepresentation Y.field)
          hAB)
        e x).1) =
      (e x).1 := by
  cases hAB
  rfl
theorem

addSubgroupCongr_apply_val

L114 ↗
GlobalClassFieldTheory.Reciprocity.addSubgroupCongr_apply_val
Documentation

Changing only the bundled subgroup of an additive subgroup element does not change its value in the ambient group.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem addSubgroupCongr_apply_val
    {A : Type} [AddGroup A]
    {H K : AddSubgroup A}
    (h : H = K) (x : H) :
    ((AddEquiv.addSubgroupCongr h x).1 : A) = x.1
theorem

rationalIdeleClassEquivFixed_congr_apply_val

L124 ↗
GlobalClassFieldTheory.Reciprocity.rationalIdeleClassEquivFixed_congr_apply_val
Documentation

Transporting an idele class along a field equality and the corresponding algebra equivalence leaves its rational direct-limit representative fixed.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem rationalIdeleClassEquivFixed_congr_apply_val
    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
    (h : A = B)
    (e : B ≃ₐ[ℚ] A)
    (he : e.trans (IntermediateField.equivOfEq h) =
      (AlgEquiv.refl : B ≃ₐ[ℚ] B))
    (c : Additive (IdeleClassGroup B)) :
    ((rationalIdeleClassEquivFixed A)
        (MulEquiv.toAdditive (ideleClassCongr e) c)).1 =
      ((rationalIdeleClassEquivFixed B) c).1
theorem

rationalIdeleClassEquivFixed_transport_baseEquiv_val

L153 ↗
GlobalClassFieldTheory.Reciprocity.rationalIdeleClassEquivFixed_transport_baseEquiv_val
Documentation

Transporting the target intermediate field of a base-field equivalence preserves its rational fixed-part representative.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem rationalIdeleClassEquivFixed_transport_baseEquiv_val
    {T : Type} [Field T] [NumberField T]
    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
    (h : A = B) (e : T ≃ₐ[ℚ] B)
    (c : IdeleClassGroup T) :
    ((rationalIdeleClassEquivFixed A)
      (Additive.ofMul
        (ideleClassCongr (K := T) (M := A)
          (e.trans (IntermediateField.equivOfEq h).symm) c))).1 =
    ((rationalIdeleClassEquivFixed B)
      (Additive.ofMul (ideleClassCongr (K := T) (M := B) e c))).1
def

extensionQuotientMulEquivOfEq

L174 ↗
GlobalClassFieldTheory.Reciprocity.extensionQuotientMulEquivOfEq
Documentation

Equality of the lower and upper closed subgroups transports the raw extension quotient without exposing dependent rewrites to clients.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
def extensionQuotientMulEquivOfEq
    {G : Type u} [Group G] [TopologicalSpace G]
    {H H' J J' : ClosedSubgroup G}
    (hH : H = H') (hJ : J = J')
    (hJH : J.toSubgroup ≤ H.toSubgroup)
    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal] :
    (H.toSubgroup ⧸ CyclicCohomology.extensionSubgroup H J hJH) ≃*
      (H'.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H' J' hJH')
theorem

extensionQuotientMulEquivOfEq_mk

L192 ↗
GlobalClassFieldTheory.Reciprocity.extensionQuotientMulEquivOfEq_mk
Documentation

The quotient transport sends a quotient representative to the same ambient group element, rebundled in the equal lower subgroup.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem extensionQuotientMulEquivOfEq_mk
    {G : Type u} [Group G] [TopologicalSpace G]
    {H H' J J' : ClosedSubgroup G}
    (hH : H = H') (hJ : J = J')
    (hJH : J.toSubgroup ≤ H.toSubgroup)
    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal]
    (σ : H.toSubgroup) :
    extensionQuotientMulEquivOfEq hH hJ hJH hJH'
        (QuotientGroup.mk σ) =
      QuotientGroup.mk
        ((MulEquiv.subgroupCongr
          (congrArg ClosedSubgroup.toSubgroup hH)) σ)
theorem

closedSubgroupCongr_apply_val

L212 ↗
GlobalClassFieldTheory.Reciprocity.closedSubgroupCongr_apply_val
Documentation

Rebundling an element along equality of closed subgroups preserves its underlying ambient group element.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem closedSubgroupCongr_apply_val
    {G : Type u} [Group G] [TopologicalSpace G]
    {H H' : ClosedSubgroup G}
    (hH : H = H') (σ : H.toSubgroup) :
    (((MulEquiv.subgroupCongr
        (congrArg ClosedSubgroup.toSubgroup hH)) σ).1 : G) = σ.1
def

abelianizedExtensionQuotientAddEquiv_transportBase

L223 ↗
GlobalClassFieldTheory.Reciprocity.abelianizedExtensionQuotientAddEquiv_transportBase
Documentation

Rebase an abelianized extension-quotient equivalence together with its finite abstract base.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
def abelianizedExtensionQuotientAddEquiv_transportBase
    {G : Type u} [Group G] [TopologicalSpace G]
    {A B : FiniteAbstractField G}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    {X : Type} [AddGroup X]
    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
    Additive
        (Abelianization
          (Eq.mp
            (congrArg
              (fun Y : FiniteAbstractField G =>
                FiniteGaloisSubextension Y.field)
              hAB)
            P).extensionQuotient) ≃+ X := by
  cases hAB
  exact e
def

abelianizedExtensionQuotientAddEquiv_transportExtension

L243 ↗
GlobalClassFieldTheory.Reciprocity.abelianizedExtensionQuotientAddEquiv_transportExtension
Documentation

Rebase an abelianized equivalence along equality of finite Galois subextensions over a fixed abstract base.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
def abelianizedExtensionQuotientAddEquiv_transportExtension
    {G : Type u} [Group G] [TopologicalSpace G]
    {K : ClosedSubgroup G}
    {P Q : FiniteGaloisSubextension K}
    (hPQ : P = Q)
    {X : Type} [AddGroup X]
    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
    Additive (Abelianization Q.extensionQuotient) ≃+ X
def

extensionQuotientMulEquiv_transportFiniteGalois

L256 ↗
GlobalClassFieldTheory.Reciprocity.extensionQuotientMulEquiv_transportFiniteGalois
Documentation

The explicit quotient equivalence induced by rebasing a finite Galois subextension.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
def extensionQuotientMulEquiv_transportFiniteGalois
    {G : Type u} [Group G] [TopologicalSpace G]
    {A B : FiniteAbstractField G}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    {Q : FiniteGaloisSubextension B.field}
    (hPQ :
      Eq.mp
        (congrArg
          (fun Y : FiniteAbstractField G =>
            FiniteGaloisSubextension Y.field)
          hAB)
        P = Q) :
    Q.extensionQuotient ≃* P.extensionQuotient := by
  cases hAB
  cases hPQ
  exact MulEquiv.refl _
theorem

extensionQuotientMulEquiv_transportFiniteGalois_mk

L277 ↗
GlobalClassFieldTheory.Reciprocity.extensionQuotientMulEquiv_transportFiniteGalois_mk
Documentation

Quotient rebasing sends a canonical representative to the same ambient group element rebundled in the old base subgroup.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
theorem extensionQuotientMulEquiv_transportFiniteGalois_mk
    {G : Type u} [Group G] [TopologicalSpace G]
    {A B : FiniteAbstractField G}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    {Q : FiniteGaloisSubextension B.field}
    (hPQ :
      Eq.mp
        (congrArg
          (fun Y : FiniteAbstractField G =>
            FiniteGaloisSubextension Y.field)
          hAB)
        P = Q)
    (σ : B.field.toSubgroup) :
    extensionQuotientMulEquiv_transportFiniteGalois hAB P hPQ
        (Q.extensionQuotientMk σ) =
      P.extensionQuotientMk
        ((MulEquiv.subgroupCongr
          (congrArg ClosedSubgroup.toSubgroup
            (congrArg FiniteAbstractField.field hAB).symm)) σ) := by
  cases hAB
  cases hPQ
  rfl
theorem

abelianizedCanonicalEquiv_transportFiniteGalois

L303 ↗
GlobalClassFieldTheory.Reciprocity.abelianizedCanonicalEquiv_transportFiniteGalois
Documentation

Abelianization commutes with simultaneous transport of the abstract base and its finite Galois subextension.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
theorem abelianizedCanonicalEquiv_transportFiniteGalois
    {G : Type u} [Group G] [TopologicalSpace G]
    {A B : FiniteAbstractField G}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    {Q : FiniteGaloisSubextension B.field}
    (hPQ :
      Eq.mp
        (congrArg
          (fun Y : FiniteAbstractField G =>
            FiniteGaloisSubextension Y.field)
          hAB)
        P = Q)
    {X : Type} [CommGroup X]
    (e : P.extensionQuotient ≃* X) :
    abelianizedExtensionQuotientAddEquiv_transportExtension hPQ
        (abelianizedExtensionQuotientAddEquiv_transportBase
          hAB P
          (MulEquiv.toAdditive
            (e.abelianizationCongr.trans
              (Abelianization.equivOfComm : X ≃* Abelianization X).symm))) =
      MulEquiv.toAdditive
        (((extensionQuotientMulEquiv_transportFiniteGalois
            hAB P hPQ).trans e).abelianizationCongr.trans
          (Abelianization.equivOfComm : X ≃* Abelianization X).symm) := by
  cases hAB
  cases hPQ
  rfl
def

rationalFiniteNormResidueValue

L334 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue
Documentation

Package the entire rational finite norm-residue evaluation so that its dependent base, extension, quotient and comparison maps move together.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
def rationalFiniteNormResidueValue
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteGaloisSubextension K.field)
    {C X : Type} [AddGroup C] [AddGroup X]
    (eIdele : C ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
    (c : C) : X
theorem

rationalFiniteNormResidueValue_transportBase

L361 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_transportBase
Documentation

The packaged norm-residue value is invariant under rebasing the finite abstract field together with all dependent data.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · full declaration
theorem rationalFiniteNormResidueValue_transportBase
    {A B : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
    (hAB : A = B)
    (P : FiniteGaloisSubextension A.field)
    {C X : Type} [AddGroup C] [AddGroup X]
    (eIdele : C ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
    (c : C) :
    rationalFiniteNormResidueValue A P eIdele eGalois c =
      rationalFiniteNormResidueValue B
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              FiniteGaloisSubextension Y.field)
            hAB)
          P)
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              C ≃+ ambientFixedAddSubgroup
                rationalIdeleClassRepresentation Y.field)
            hAB)
          eIdele)
        (abelianizedExtensionQuotientAddEquiv_transportBase
          hAB P eGalois)
        c := by
  cases hAB
  rfl
theorem

rationalFiniteNormResidueValue_transportExtension

L396 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_transportExtension
Documentation

The packaged norm-residue value is invariant under equality of the finite Galois subextension.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem rationalFiniteNormResidueValue_transportExtension
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    {P Q : FiniteGaloisSubextension K.field}
    (hPQ : P = Q)
    {C X : Type} [AddGroup C] [AddGroup X]
    (eIdele : C ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
    (c : C) :
    rationalFiniteNormResidueValue K P eIdele eGalois c =
      rationalFiniteNormResidueValue K Q eIdele
        (abelianizedExtensionQuotientAddEquiv_transportExtension
          hPQ eGalois)
        c
theorem

abelianizationCongrToComm_apply

L416 ↗
GlobalClassFieldTheory.Reciprocity.abelianizationCongrToComm_apply
Documentation

Evaluation of the canonical abelianization comparison induced by a multiplicative equivalence into a commutative group.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem abelianizationCongrToComm_apply
    {Q R : Type*} [Group Q] [CommGroup R]
    (e : Q ≃* R) (q : Q) :
    MulEquiv.toAdditive
        (e.abelianizationCongr.trans
          (Abelianization.equivOfComm : R ≃* Abelianization R).symm)
        (Additive.ofMul (Abelianization.of q)) =
      Additive.ofMul (e q)
theorem

commutativeAbelianizationEquiv_apply

L434 ↗
GlobalClassFieldTheory.Reciprocity.commutativeAbelianizationEquiv_apply
Documentation

Evaluation of the canonical quotient from the abelianization of a commutative group.

Source context 14 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
Lean source · statement
theorem commutativeAbelianizationEquiv_apply
    {Q R : Type*} [CommGroup Q] [Group R]
    (e : Q ≃* R) (q : Q) :
    MulEquiv.toAdditive
        ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm.trans e)
        (Additive.ofMul (Abelianization.of q)) =
      Additive.ofMul (e q)
instance

naturalityAbstractFixedFieldBaseQuotientFinite

L455 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractFixedFieldBaseQuotientFinite
Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
Lean source · statement
instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field))
instance

naturalityAbstractFixedFieldRelativeQuotientFinite

L465 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractFixedFieldRelativeQuotientFinite
Source context 17 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
Lean source · statement
instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below)
instance

naturalityAbstractFixedFieldFiniteDimensional

L471 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractFixedFieldFiniteDimensional
Source context 18 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
Lean source · statement
instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
instance

naturalityAbstractRelativeFixedFieldFiniteDimensional

L478 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractRelativeFixedFieldFiniteDimensional
Source context 19 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
Lean source · statement
instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below)
instance

naturalityAbstractFixedFieldRelativeScalarTower

L488 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractFixedFieldRelativeScalarTower
Source context 20 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
Lean source · statement
instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below)
instance

naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional

L495 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional
Source context 21 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
Lean source · statement
instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below)
instance

naturalityAbstractFixedFieldNumberField

L505 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractFixedFieldNumberField
Source context 22 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
Lean source · statement
instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
instance

naturalityAbstractRelativeFixedFieldNumberField

L511 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractRelativeFixedFieldNumberField
Source context 23 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
Lean source · statement
instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below)
instance

naturalityAbstractRelativeFixedFieldIsGalois

L523 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractRelativeFixedFieldIsGalois
Documentation

Use the same direct fixed-field Galois witness as the intrinsic norm-residue construction. This prevents the dependent Galois-group type from being synthesized through a second IsAbelianGalois instance path.

Source context 24 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
Lean source · statement
instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below)
instance

naturalityAbstractRelativeFixedFieldIsAbelianGalois

L533 ↗
GlobalClassFieldTheory.Reciprocity.naturalityAbstractRelativeFixedFieldIsAbelianGalois
Source context 25 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
Lean source · statement
instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
theorem

numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion

L543 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion
Documentation

The lower subgroup obtained from the canonical inclusion of an abstract fixed-field tower is the original lower closed subgroup.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    numberFieldEmbeddedBaseSubgroup F E j = H.field := by
  dsimp only
  have hi :
      numberFieldEmbeddedLowerEmbedding
          (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
          (abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) P.below)
          ((abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) =
        (abstractFixedField
          ℚ (SeparableClosure ℚ) H.field).val := by
    ext x
    rfl
  have hRange :
      (numberFieldEmbeddedLowerEmbedding
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below)
        ((abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ)).fieldRange =
        abstractFixedField ℚ (SeparableClosure ℚ) H.field := by
    ext x
    constructor
    · rintro ⟨y, rfl⟩
      change
        numberFieldEmbeddedLowerEmbedding
            (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
            (abstractRelativeFixedField
              ℚ (SeparableClosure ℚ) P.below)
            ((abstractRelativeFixedField
              ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) y ∈
          abstractFixedField ℚ (SeparableClosure ℚ) H.field
      rw [hi]
      exact y.property
    · intro hx
      refine ⟨⟨x, hx⟩, ?_⟩
      rw [hi]
      rfl
  rw [numberFieldEmbeddedBaseSubgroup, hRange]
  exact
    closedFixingSubgroup_abstractFixedField_eq
      ℚ (SeparableClosure ℚ) H.field
theorem

numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion

L600 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion
Documentation

The upper subgroup obtained from the canonical inclusion of an abstract fixed-field tower is the original upper closed subgroup.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    numberFieldEmbeddedTopSubgroup F E j = P.field := by
  dsimp only
  rw [numberFieldEmbeddedTopSubgroup]
  have hjRangeSelf :
      ((abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below).restrictScalars ℚ := by
    ext x
    constructor
    · rintro ⟨y, rfl⟩
      exact y.property
    · intro hx
      exact ⟨⟨x, hx⟩, rfl⟩
  have hjRange :
      ((abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
        abstractFixedField ℚ (SeparableClosure ℚ) P.field := by
    exact hjRangeSelf.trans
      (IntermediateField.extendScalars_restrictScalars
        (abstractFixedField_le
          ℚ (SeparableClosure ℚ) P.below))
  rw [hjRange]
  exact
    closedFixingSubgroup_abstractFixedField_eq
      ℚ (SeparableClosure ℚ) P.field
theorem

rationalFiniteNormResidueValue_transportToAbstractExtension

L643 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_transportToAbstractExtension
Documentation

Transport a packaged rational norm-residue value directly from an equal abstract base to the finite Galois extension underlying P. Keeping the two dependent transports in their own declaration prevents their elaboration cost from accumulating in the main fixed-field comparison theorem.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem rationalFiniteNormResidueValue_transportToAbstractExtension
    {A : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
    (hAH : A = H)
    (L : FiniteGaloisSubextension A.field)
    (hLP :
      Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              FiniteGaloisSubextension Y.field)
            hAH)
          L =
        P.toFiniteGaloisExtension)
    {C X : Type} [AddGroup C] [AddGroup X]
    (eIdele : C ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
    (c : C) :
    rationalFiniteNormResidueValue A L eIdele eGalois c =
      rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              C ≃+ ambientFixedAddSubgroup
                rationalIdeleClassRepresentation Y.field)
            hAH)
          eIdele)
        (abelianizedExtensionQuotientAddEquiv_transportExtension hLP
          (abelianizedExtensionQuotientAddEquiv_transportBase
            hAH L eGalois))
        c := by
  calc
    _ = rationalFiniteNormResidueValue H
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              FiniteGaloisSubextension Y.field)
            hAH)
          L)
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              C ≃+ ambientFixedAddSubgroup
                rationalIdeleClassRepresentation Y.field)
            hAH)
          eIdele)
        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
        c :=
      rationalFiniteNormResidueValue_transportBase
        (A := A) (B := H) (C := C) (X := X)
        hAH L eIdele eGalois c
    _ = _ :=
      rationalFiniteNormResidueValue_transportExtension
        (K := H)
        (P := Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              FiniteGaloisSubextension Y.field)
            hAH)
          L)
        (Q := P.toFiniteGaloisExtension) (C := C) (X := X)
        hLP
        (Eq.mp
          (congrArg
            (fun Y : FiniteAbstractField
                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
              C ≃+ ambientFixedAddSubgroup
                rationalIdeleClassRepresentation Y.field)
            hAH)
          eIdele)
        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
        c
theorem

rationalFiniteNormResidueValue_abstractFixedField_eq_ambient

L723 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
Documentation

The packaged value at the literal fixed-field realization is the ambient fixed-part norm-residue homomorphism evaluated at the same idele class.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
    (c : IdeleClassGroup
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
    rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
        (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
        (Additive.ofMul c) =
      ambientFixedGlobalNormResidueAddMonoidHom H P
        (rationalAbstractFixedFieldIdeleClassEquivFixed
          H.field (Additive.ofMul c))
theorem

rationalFiniteNormResidueValue_abstractFixedField_apply

L758 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_abstractFixedField_apply
Documentation

The packaged norm-residue value for the literal fixed-field realization is the intrinsic abstract fixed-field norm-residue value.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem rationalFiniteNormResidueValue_abstractFixedField_apply
    (c : IdeleClassGroup
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
    Additive.toMul
        (rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
          (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
          (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
            H P)
          (Additive.ofMul c)) =
      abstractFixedFieldGlobalNormResidueMonoidHom H P c
theorem

numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion

L788 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion
Documentation

The finite abstract field reconstructed from the literal fixed-field inclusion is the original packaged abstract field. Keeping this structure equality separate avoids repeatedly rebuilding all of its proof fields.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    numberFieldEmbeddedFiniteAbstractField F E j = H := by
  dsimp only
  exact FiniteAbstractField.eq_of_field_eq _ _
    (numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P)
theorem

numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply

L809 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply
Documentation

Pointwise specification of the idele-class comparison after transporting the explicitly embedded abstract field to the canonical packaged field. The transport is kept at the value boundary, so clients never compare the two dependent additive equivalences themselves.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    let HEmbedded :=
      numberFieldEmbeddedFiniteAbstractField F E j
    let hHEmbedded : HEmbedded = H :=
      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
    Eq.mp
        (congrArg
          (fun X : FiniteAbstractField
              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
            Additive (IdeleClassGroup F) ≃+
              ambientFixedAddSubgroup
                rationalIdeleClassRepresentation X.field)
          hHEmbedded)
        (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c =
      rationalAbstractFixedFieldIdeleClassEquivFixed H.field c := by
  dsimp only
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) H.field
  let E :=
    abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below
  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
    E.val.restrictScalars ℚ
  have hBase :
      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
  have hHEmbedded :
      numberFieldEmbeddedFiniteAbstractField F E j = H :=
    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
  have hFixedBase :
      abstractFixedField ℚ (SeparableClosure ℚ)
          (numberFieldEmbeddedBaseSubgroup F E j) = F :=
    congrArg
      (abstractFixedField ℚ (SeparableClosure ℚ)) hBase
  let eBase : F ≃ₐ[ℚ] F :=
    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j).trans
      (IntermediateField.equivOfEq hFixedBase)
  have heBase :
      eBase = (AlgEquiv.refl : F ≃ₐ[ℚ] F) := by
    apply AlgEquiv.ext
    intro x
    apply Subtype.ext
    change x.1 = x.1
    rfl
  let hEmbeddedQuotientFinite :=
    numberFieldEmbeddedAbsoluteQuotientFinite F E j
  let hEmbeddedFixedFiniteDimensional :=
    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional F E j
  apply Subtype.ext
  rw [rationalAmbientFixedAddEquiv_transport_apply_val
    hHEmbedded
    (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c]
  dsimp only [numberFieldEmbeddedIdeleClassEquivAmbientFixed]
  simp only [AddEquiv.trans_apply]
  change
    ((rationalIdeleClassEquivFixed
        (abstractFixedField ℚ (SeparableClosure ℚ)
          (numberFieldEmbeddedBaseSubgroup F E j)))
      (MulEquiv.toAdditive
        (ideleClassCongr
          (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)) c)).1 =
      ((rationalIdeleClassEquivFixed F) c).1
  exact rationalIdeleClassEquivFixed_congr_apply_val
    hFixedBase
    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)
    heBase c
theorem

numberFieldEmbeddedFiniteGaloisSubextension_transport_eq

L894 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedFiniteGaloisSubextension_transport_eq
Documentation

Transporting the finite Galois subextension reconstructed from the literal fixed-field inclusion recovers the canonical subextension packaged by P. This is the sole dependent structure equality used by the later quotient comparisons.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    let HEmbedded :=
      numberFieldEmbeddedFiniteAbstractField F E j
    let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
      numberFieldEmbeddedFiniteGaloisSubextension F E j
    let hHEmbedded : HEmbedded = H :=
      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
    Eq.mp
        (congrArg
          (fun X : FiniteAbstractField
              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
            FiniteGaloisSubextension X.field)
          hHEmbedded)
        PEmbedded =
      P.toFiniteGaloisExtension := by
  dsimp only
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) H.field
  let E :=
    abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below
  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
    E.val.restrictScalars ℚ
  let HEmbedded :=
    numberFieldEmbeddedFiniteAbstractField F E j
  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
    numberFieldEmbeddedFiniteGaloisSubextension F E j
  have hHEmbedded : HEmbedded = H :=
    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
  exact finiteGaloisSubextension_transport_eq_of_field_eq
    hHEmbedded PEmbedded P.toFiniteGaloisExtension
    (numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P)
def

abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv

L941 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv
Documentation

Opaque endpoint for the extension-quotient comparison supplied by the literal embedding. Its domain is already the canonical quotient of P, so no client has to reconstruct the two subgroup transports.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    P.toFiniteGaloisExtension.extensionQuotient ≃*
      Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv

L984 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv
Documentation

Opaque canonical endpoint for the same quotient, obtained directly from the abstract fixed-field realization.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    P.toFiniteGaloisExtension.extensionQuotient ≃*
      Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionEmbeddedExtensionQuotientValue

L1001 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
Documentation

Pointwise opaque endpoint of the embedded quotient equivalence.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    Gal(
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) /
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionCanonicalExtensionQuotientValue

L1014 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalExtensionQuotientValue
Documentation

Pointwise opaque endpoint of the canonical quotient equivalence.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionCanonicalExtensionQuotientValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    Gal(
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) /
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal

L1027 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal
Documentation

Fully applied ambient value of the embedded quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ
def

abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal

L1039 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal
Documentation

Fully applied ambient value of the canonical quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ
def

abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue

L1052 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue
Documentation

The ambient Galois value attached to a representative of the canonical quotient, packaged behind a literal result type.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup) :
    Gal(
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) /
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal

L1091 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
Documentation

Fully applied ambient value of the packaged representative endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ
def

abstractFixedFieldInclusionRebasedAutomorphismApplyVal

L1104 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionRebasedAutomorphismApplyVal
Documentation

Ambient action of the representative after rebundling it in the embedded base subgroup.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionRebasedAutomorphismApplyVal
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ
theorem

abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk

L1130 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
Documentation

The embedded quotient endpoint sends a canonical representative to the packaged ambient value above.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup) :
    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
      abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue H P σ
theorem

abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased

L1176 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
Documentation

The packaged ambient endpoint evaluates to the action of the rebundled representative.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
      abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x
theorem

abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq

L1225 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq
Documentation

Rebundling the representative does not change its action in the ambient separable closure.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :
    abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x =
      σ.1.1 (x : SeparableClosure ℚ)
theorem

abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply

L1257 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
Documentation

The packaged ambient representative acts by the original automorphism on the underlying separable-closure value.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :
    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
      σ.1.1 (x : SeparableClosure ℚ)
theorem

abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val

L1274 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
Documentation

Evaluation of the embedded quotient endpoint on a canonical quotient representative, stated only in the ambient separable closure.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :
    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal H P
        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
      σ.1.1 (x : SeparableClosure ℚ)
theorem

abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val

L1298 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
Documentation

Evaluation of the canonical abstract quotient endpoint on a quotient representative, again exposed only through its ambient value.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup)
    (x : abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :
    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal H P
        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
      σ.1.1 (x : SeparableClosure ℚ)
theorem

abstractFixedFieldInclusionExtensionQuotientEquiv_apply

L1323 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionExtensionQuotientEquiv_apply
Documentation

The embedded and canonical quotient endpoints agree on each quotient class. This pointwise boundary is intentionally weaker than equality of the dependent MulEquiv structures.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionExtensionQuotientEquiv_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q =
      abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q
def

abstractFixedFieldInclusionTransportedExtensionQuotientEquiv

L1348 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedExtensionQuotientEquiv
Documentation

Opaque quotient endpoint obtained by transporting the explicitly embedded finite Galois subextension back to the canonical package.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    P.toFiniteGaloisExtension.extensionQuotient ≃*
      Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
def

abstractFixedFieldInclusionTransportedExtensionQuotientValue

L1387 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedExtensionQuotientValue
Documentation

Pointwise opaque endpoint of the transported quotient equivalence.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedExtensionQuotientValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    Gal(
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) /
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
theorem

abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk

L1401 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
Documentation

Transporting a canonical representative of the embedded finite Galois package preserves its Galois value.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (σ : H.field.toSubgroup) :
    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P
        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
        (P.toFiniteGaloisExtension.extensionQuotientMk σ)
theorem

abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply

L1477 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
Documentation

Transporting the embedded finite Galois package preserves the value of its extension-quotient comparison.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P q =
      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q
def

abstractFixedFieldInclusionTransportedAbelianizedEquiv

L1496 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedAbelianizedEquiv
Documentation

Opaque abelianized equivalence obtained by transporting the explicitly embedded finite Galois package to the canonical one.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedAbelianizedEquiv
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    Additive
        (Abelianization
          P.toFiniteGaloisExtension.extensionQuotient) ≃+
      Additive
        (Gal(
          (abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) P.below) /
          (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv

L1541 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
Documentation

Canonical abelianization comparison built from the already transported opaque extension-quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    Additive
        (Abelianization
          P.toFiniteGaloisExtension.extensionQuotient) ≃+
      Additive
        (Gal(
          (abstractRelativeFixedField
            ℚ (SeparableClosure ℚ) P.below) /
          (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionTransportedAbelianizedValue

L1566 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedAbelianizedValue
Documentation

Pointwise opaque value of the transported abelianized equivalence.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedAbelianizedValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue

L1582 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
Documentation

Pointwise opaque value of the canonical abelianization comparison built from the transported quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionCanonicalAbelianizedValue

L1598 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalAbelianizedValue
Documentation

Pointwise opaque value of the intrinsic abstract fixed-field abelianization comparison.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionCanonicalAbelianizedValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
theorem

abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical

L1614 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
Documentation

The transported abelianized endpoint agrees pointwise with the canonical abelianization comparison built from the transported quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
      abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
        H P z
theorem

abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of

L1665 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
Documentation

On an abelianization representative, the transported canonical endpoint is the additive value of the transported quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P
        (Additive.ofMul (Abelianization.of q)) =
      Additive.ofMul
        (abstractFixedFieldInclusionTransportedExtensionQuotientValue
          H P q)
theorem

abstractFixedFieldInclusionCanonicalAbelianizedValue_of

L1685 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalAbelianizedValue_of
Documentation

On the same representative, the intrinsic fixed-field endpoint is the additive value of the canonical quotient endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionCanonicalAbelianizedValue_of
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (q : P.toFiniteGaloisExtension.extensionQuotient) :
    abstractFixedFieldInclusionCanonicalAbelianizedValue H P
        (Additive.ofMul (Abelianization.of q)) =
      Additive.ofMul
        (abstractFixedFieldInclusionCanonicalExtensionQuotientValue
          H P q)
theorem

abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq

L1716 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
Documentation

The canonical abelianization comparison built after transport agrees pointwise with the intrinsic abstract fixed-field comparison.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P z =
      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z
theorem

abstractFixedFieldInclusionTransportedAbelianizedValue_eq

L1753 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedAbelianizedValue_eq
Documentation

The transported abelianized equivalence agrees pointwise with the intrinsic abstract fixed-field Galois comparison.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedAbelianizedValue_eq
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (z : Additive
      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z
def

abstractFixedFieldInclusionEmbeddedNormResidueValue

L1769 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedNormResidueValue
Documentation

Opaque packaged norm-residue value before transporting the explicitly embedded abstract field and finite Galois package.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionEmbeddedNormResidueValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionTransportedNormResidueValue

L1802 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedNormResidueValue
Documentation

Opaque packaged value after transporting both dependent structures to the canonical H and P endpoints.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionTransportedNormResidueValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
def

abstractFixedFieldInclusionCanonicalNormResidueValue

L1847 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionCanonicalNormResidueValue
Documentation

Opaque intrinsic packaged norm-residue value at the canonical endpoints.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
def
    abstractFixedFieldInclusionCanonicalNormResidueValue
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    Additive
      (Gal(
        (abstractRelativeFixedField
          ℚ (SeparableClosure ℚ) P.below) /
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)))
theorem

abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported

L1867 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
Documentation

Simultaneous transport of the embedded abstract field and finite Galois package sends the embedded norm-residue value to the transported endpoint.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
      abstractFixedFieldInclusionTransportedNormResidueValue H P c
theorem

rationalFiniteNormResidueValue_congr_apply

L1921 ↗
GlobalClassFieldTheory.Reciprocity.rationalFiniteNormResidueValue_congr_apply
Documentation

A packaged finite norm-residue value depends only on the value of its idele comparison at the chosen input and the value of its Galois comparison at the resulting norm class.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem rationalFiniteNormResidueValue_congr_apply
    (K : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (L : FiniteGaloisSubextension K.field)
    {C X : Type} [AddGroup C] [AddGroup X]
    (eIdele eIdele' : C ≃+
      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
    (eGalois eGalois' :
      Additive (Abelianization L.extensionQuotient) ≃+ X)
    (c : C)
    (heIdele : eIdele c = eIdele' c)
    (heGalois : ∀ z, eGalois z = eGalois' z) :
    rationalFiniteNormResidueValue K L eIdele eGalois c =
      rationalFiniteNormResidueValue K L eIdele' eGalois' c
theorem

abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical

L1941 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
Documentation

The transported packaged value is the intrinsic canonical packaged value; only the idele input and the eventual abelianized value are compared.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    abstractFixedFieldInclusionTransportedNormResidueValue H P c =
      abstractFixedFieldInclusionCanonicalNormResidueValue H P c
theorem

abstractFixedFieldInclusionEmbeddedNormResidueValue_eq

L1993 ↗
GlobalClassFieldTheory.Reciprocity.abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
Documentation

The explicitly embedded and intrinsic packaged norm-residue values agree.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · statement
theorem
    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
    (c : Additive
      (IdeleClassGroup
        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
      abstractFixedFieldInclusionCanonicalNormResidueValue H P c
theorem

globalNormResidueMonoidHomOfEmbedding_abstractFixedFieldInclusion

L2012 ↗
GlobalClassFieldTheory.Reciprocity.globalNormResidueMonoidHomOfEmbedding_abstractFixedFieldInclusion
Documentation

For the literal fixed fields attached to an abstract finite abelian extension, the norm-residue map obtained from their canonical inclusion in the rational separable closure is the intrinsic fixed-field norm-residue map.

Source context 26 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
section AbstractFixedFieldInclusion
variable
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field)
local instance naturalityAbstractFixedFieldBaseQuotientFinite :
    Finite
      ((baseField
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (baseField
            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
          H.field (le_baseField H.field)) :=
  H.finite
local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
    Finite
      (H.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
  P.finite
noncomputable local instance
    naturalityAbstractFixedFieldFiniteDimensional :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) H.field H.finite
noncomputable local instance
    naturalityAbstractRelativeFixedFieldFiniteDimensional :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below H.finite P.finite
local instance naturalityAbstractFixedFieldRelativeScalarTower :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
    FiniteDimensional ℚ
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
noncomputable local instance naturalityAbstractFixedFieldNumberField :
    NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
  NumberField.of_module_finite ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
noncomputable local instance
    naturalityAbstractRelativeFixedFieldNumberField :
    NumberField
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below)
/-- Use the same direct fixed-field Galois witness as the intrinsic
norm-residue construction.  This prevents the dependent Galois-group type
from being synthesized through a second `IsAbelianGalois` instance path. -/
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsGalois :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    H.field P.field P.below P.normal
noncomputable local instance
    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
    IsAbelianGalois
      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
      (abstractRelativeFixedField
      ℚ (SeparableClosure ℚ) P.below) :=
  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
Lean source · full declaration
theorem
    globalNormResidueMonoidHomOfEmbedding_abstractFixedFieldInclusion
    (H : FiniteAbstractField
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (P : FiniteAbelianSubextension H.field) :
    let F :=
      abstractFixedField ℚ (SeparableClosure ℚ) H.field
    let E :=
      abstractRelativeFixedField
        ℚ (SeparableClosure ℚ) P.below
    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
      E.val.restrictScalars ℚ
    globalNormResidueMonoidHomOfEmbedding F E j =
      abstractFixedFieldGlobalNormResidueMonoidHom H P := by
  dsimp only
  apply MonoidHom.ext
  intro c
  rw [globalNormResidueMonoidHomOfEmbedding_apply]
  change
    Additive.toMul
        (abstractFixedFieldInclusionEmbeddedNormResidueValue
          H P (Additive.ofMul c)) =
      abstractFixedFieldGlobalNormResidueMonoidHom H P c
  calc
    _ = Additive.toMul
        (abstractFixedFieldInclusionCanonicalNormResidueValue
          H P (Additive.ofMul c)) :=
      congrArg Additive.toMul
        (abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
          H P (Additive.ofMul c))
    _ = _ := by
      simpa only [abstractFixedFieldInclusionCanonicalNormResidueValue] using
        (rationalFiniteNormResidueValue_abstractFixedField_apply
          (H := H) (P := P) c)
theorem

numberFieldEmbeddedBaseSubgroup_baseChange_eq

L2064 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseSubgroup_baseChange_eq
Documentation

Rebracketing the compatible tower does not change its embedded lower fixing subgroup.

Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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 · statement
theorem numberFieldEmbeddedBaseSubgroup_baseChange_eq
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    numberFieldEmbeddedBaseSubgroup K K'
        (numberFieldEmbeddedLowerEmbedding K' L' j) =
      numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))
theorem

numberFieldEmbeddedTopSubgroup_baseChange_eq

L2086 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedTopSubgroup_baseChange_eq
Documentation

The top subgroup of the rebracketed base-change tower is the embedded fixing subgroup of the intermediate field.

Source context 17 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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]
    [Algebra K K'] [Algebra K L'] [IsScalarTower K K' L'] in
Lean source · statement
theorem numberFieldEmbeddedTopSubgroup_baseChange_eq
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    numberFieldEmbeddedTopSubgroup K K'
        (numberFieldEmbeddedLowerEmbedding K' L' j) =
      numberFieldEmbeddedBaseSubgroup K' L' j
def

numberFieldEmbeddedBaseChangeBaseEquiv

L2095 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeBaseEquiv
Documentation

Reidentify the two presentations of the embedded lower fixing subgroup without transporting dependent subgroup data through an equality.

Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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 · statement
def numberFieldEmbeddedBaseChangeBaseEquiv
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (numberFieldEmbeddedBaseSubgroup K K'
        (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ≃*
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup
theorem

numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq

L2107 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
Documentation

Under the identity equivalence of the two lower fixing subgroups, the relative subgroup for the base change is exactly the target presentation.

Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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 · statement
theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
        (numberFieldEmbeddedBaseSubgroup K K'
          (numberFieldEmbeddedLowerEmbedding K' L' j))
        (numberFieldEmbeddedTopSubgroup K K'
          (numberFieldEmbeddedLowerEmbedding K' L' j))
        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K K'
          (numberFieldEmbeddedLowerEmbedding K' L' j))).map
          (numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j).toMonoidHom =
      CyclicCohomology.extensionSubgroup
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))
        (numberFieldEmbeddedBaseSubgroup K' L' j)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
theorem

numberFieldEmbeddedBaseChangeExtensionSubgroup_normal

L2157 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeExtensionSubgroup_normal
Documentation

Normality of the relative subgroup between the two embedded base fields in a finite Galois base change.

Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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 · statement
theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_normal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal
instance

numberFieldEmbeddedBaseChangeExtensionSubgroupNormal

L2174 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
Source context 16 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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 · statement
instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal
theorem

numberFieldEmbeddedBaseChangeExtensionQuotient_finite

L2187 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeExtensionQuotient_finite
Documentation

Finiteness of the relative quotient between the two embedded base fields in a finite Galois base change.

Source context 17 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
Lean source · statement
theorem numberFieldEmbeddedBaseChangeExtensionQuotient_finite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeExtensionQuotientFinite

L2247 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeExtensionQuotientFinite
Source context 17 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional

L2264 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
Documentation

Reuse the canonical absolute fixed-field witness for the lower embedded tower. The base-change relative witness below needs this exact instance path when forming the absolute finite-dimensional tower.

Source context 18 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
instance

numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional

L2274 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
Source context 19 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower

L2294 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
Source context 20 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional

L2306 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
Source context 21 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField

L2320 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
Source context 22 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
instance

numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois

L2331 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
Source context 23 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
Lean source · statement
instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
theorem

numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction

L2352 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
Documentation

In one common rational-separable-closure realization, the canonical quotient-to-Galois comparisons intertwine abstract restriction with ordinary restriction of the actual number-field automorphisms.

Source context 24 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j)
Lean source · full declaration
theorem
    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
    [FiniteDimensional K L] [IsAbelianGalois K L]
    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
    (z :
      Abelianization
        (numberFieldEmbeddedFiniteGaloisSubextension
          K' L' j).extensionQuotient) :
    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
      j.comp (IsScalarTower.toAlgHom ℚ L L')
    let H :=
      numberFieldEmbeddedBaseSubgroup K L jLower
    let H' :=
      numberFieldEmbeddedBaseSubgroup K' L' j
    let J :=
      numberFieldEmbeddedTopSubgroup K L jLower
    let J' :=
      numberFieldEmbeddedTopSubgroup K' L' j
    let hH'H :=
      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
    let hJ'J :=
      numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
    let hJH :=
      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
    let hJ'H' :=
      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
    letI _ :
        (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
      numberFieldEmbeddedExtensionSubgroup_normal K L jLower
    letI _ :
        (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
      numberFieldEmbeddedExtensionSubgroup_normal K' L' j
    ((AlgEquiv.restrictNormalHom L).comp
        (AlgEquiv.restrictScalarsHom K))
        (Additive.toMul
          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
            K' L' j (Additive.ofMul z))) =
      Additive.toMul
        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
          K L jLower
          (MonoidHom.toAdditive
            (normResidueNaturalityAbelianizedRestriction
              H H' J J'
              hJH hJ'H'
              hH'H hJ'J)
            (Additive.ofMul z))) := by
  dsimp only
  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
    j.comp (IsScalarTower.toAlgHom ℚ L L')
  let H :=
    numberFieldEmbeddedBaseSubgroup K L jLower
  let H' :=
    numberFieldEmbeddedBaseSubgroup K' L' j
  let J :=
    numberFieldEmbeddedTopSubgroup K L jLower
  let J' :=
    numberFieldEmbeddedTopSubgroup K' L' j
  let hH'H :=
    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
  let hJ'J :=
    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
  let hJH :=
    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
      K L jLower
  let hJ'H' :=
    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
      K' L' j
  let hLowerNormal :
      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
  let hUpperNormal :
      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
  let qLower :=
    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
      K L jLower
  let qUpper :=
    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
      K' L' j
  let qLowerRaw :
      (H.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H J hJH) ≃*
        Gal(L / K) := by
    exact
      { qLower.toEquiv with
        map_mul' := fun x y => qLower.map_mul x y }
  let qUpperRaw :
      (H'.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H' J' hJ'H') ≃*
        Gal(L' / K') := by
    exact
      { qUpper.toEquiv with
        map_mul' := fun x y => qUpper.map_mul x y }
  let restrictActual :
      Gal(L' / K') →* Gal(L / K) :=
    (AlgEquiv.restrictNormalHom L).comp
      (AlgEquiv.restrictScalarsHom K)
  obtain ⟨q, rfl⟩ :=
    QuotientGroup.mk_surjective z
  obtain ⟨σ, rfl⟩ :=
    (numberFieldEmbeddedFiniteGaloisSubextension
      K' L' j).extensionQuotientMk_surjective q
  change
    restrictActual
        ((Abelianization.equivOfComm (H := Gal(L' / K'))).symm
          (qUpperRaw.abelianizationCongr
            (Abelianization.of (QuotientGroup.mk σ)))) =
      (Abelianization.equivOfComm (H := Gal(L / K))).symm
        (qLowerRaw.abelianizationCongr
          (normResidueNaturalityAbelianizedRestriction
            H H' J J' hJH hJ'H' hH'H hJ'J
            (Abelianization.of (QuotientGroup.mk σ))))
  rw [normResidueNaturalityAbelianizedRestriction_of_mk,
    abelianizationCongr_of, abelianizationCongr_of]
  change
    restrictActual (qUpperRaw (QuotientGroup.mk σ)) =
      qLowerRaw
        (QuotientGroup.mk (Subgroup.inclusion hH'H σ))
  apply AlgEquiv.ext
  intro x
  apply jLower.injective
  let hUpperAlgebra : Algebra K' (SeparableClosure ℚ) :=
    numberFieldEmbeddedSeparableClosureAlgebra K' L' j
  let eUpper :=
    numberFieldEmbeddedSeparableClosureEquiv K' L' j
  let hLowerAlgebra : Algebra K (SeparableClosure ℚ) :=
    numberFieldEmbeddedSeparableClosureAlgebra K L jLower
  let eLower :=
    numberFieldEmbeddedSeparableClosureEquiv K L jLower
  calc
    jLower
        (restrictActual
          (qUpperRaw (QuotientGroup.mk σ)) x) =
        j
          ((qUpperRaw (QuotientGroup.mk σ))
            (algebraMap L L' x)) := by
      exact congrArg j
        (AlgEquiv.restrictNormal_commutes
          ((AlgEquiv.restrictScalarsHom K)
            (qUpperRaw (QuotientGroup.mk σ)))
          L x)
    _ = σ.1.1
        (j (algebraMap L L' x)) := by
      exact
        ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
          ℚ K' L' j eUpper σ (algebraMap L L' x)
    _ = (Subgroup.inclusion hH'H σ).1.1
        (jLower x) := rfl
    _ = jLower
        (qLowerRaw
          (QuotientGroup.mk
            (Subgroup.inclusion hH'H σ)) x) := by
      exact
        (ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
          ℚ K L jLower eLower
          (Subgroup.inclusion hH'H σ) x).symm
theorem

numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm

L2513 ↗
GlobalClassFieldTheory.Reciprocity.numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
Documentation

In a compatible common embedding, the fixed-part relative norm between two (Galois-related) base fields is the genuine ordinary idele-class norm.

Source context 24 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j)
Lean source · full declaration
theorem numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
    (c : IdeleClassGroup K') :
    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
      j.comp (IsScalarTower.toAlgHom ℚ L L')
    let H :=
      numberFieldEmbeddedBaseSubgroup K L jLower
    let H' :=
      numberFieldEmbeddedBaseSubgroup K' L' j
    let hH'H :=
      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
    letI _ : Finite
        (H.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H H' hH'H) :=
      numberFieldEmbeddedBaseChangeExtensionQuotientFinite
        K K' L L' j
    relativeNorm rationalIdeleClassRepresentation H H' hH'H
        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
          K' L' j (Additive.ofMul c)) =
      numberFieldEmbeddedIdeleClassEquivAmbientFixed
        K L jLower
        (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
  intro jLower H H'
  let hH'H :=
    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
  let hnormal :=
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j
  let F :=
    abstractFixedField ℚ (SeparableClosure ℚ) H
  let E :=
    abstractRelativeFixedField ℚ (SeparableClosure ℚ) hH'H
  let _ :
      (CyclicCohomology.extensionSubgroup H H' hH'H).Normal :=
    hnormal
  let _ :
      Finite
        (H.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H H' hH'H) :=
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
      K K' L L' j
  let _ :
      Finite
        ((baseField
          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
          CyclicCohomology.extensionSubgroup
            (baseField
              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
            H (le_baseField H)) :=
    numberFieldEmbeddedAbsoluteQuotientFinite K L jLower
  let _ : NumberField F :=
    numberFieldEmbeddedAbstractFixedFieldNumberField K L jLower
  let _ : NumberField E :=
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
      K K' L L' j
  let _ : FiniteDimensional ℚ (E.restrictScalars ℚ) := by
    change FiniteDimensional ℚ E
    exact
      numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
        K K' L L' j
  let _ : NumberField
      (abstractFixedField ℚ (SeparableClosure ℚ) H') :=
    numberFieldEmbeddedAbstractFixedFieldNumberField K' L' j
  have hE :
      E.restrictScalars ℚ =
        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
    IntermediateField.extendScalars_restrictScalars
      (abstractFixedField_le
        ℚ (SeparableClosure ℚ) hH'H)
  let eRel :
      E ≃ₐ[ℚ]
        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
    IntermediateField.equivOfEq hE
  let eK :
      K ≃ₐ[ℚ] F :=
    numberFieldEmbeddedAbstractBaseFieldEquiv K L jLower
  let eK'Base :
      K' ≃ₐ[ℚ]
        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
    numberFieldEmbeddedAbstractBaseFieldEquiv K' L' j
  let eK' : K' ≃ₐ[ℚ] E :=
    eK'Base.trans eRel.symm
  have hcompat (x : K) :
      eK' (algebraMap K K' x) =
        algebraMap F E (eK x) := by
    apply eRel.injective
    apply Subtype.ext
    change
      j (algebraMap K' L' (algebraMap K K' x)) =
        j (algebraMap L L' (algebraMap K L x))
    rw [← IsScalarTower.algebraMap_apply K K' L',
      ← IsScalarTower.algebraMap_apply K L L']
  have hupper :
      rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
          H H' hH'H
          (Additive.ofMul (ideleClassCongr eK' c)) =
        numberFieldEmbeddedIdeleClassEquivAmbientFixed
          K' L' j (Additive.ofMul c) := by
    apply Subtype.ext
    change
      ((rationalIdeleClassEquivFixed (E.restrictScalars ℚ))
          (Additive.ofMul (ideleClassCongr eK' c))).1 =
        ((rationalIdeleClassEquivFixed
            (abstractFixedField ℚ (SeparableClosure ℚ) H'))
          (Additive.ofMul (ideleClassCongr eK'Base c))).1
    exact rationalIdeleClassEquivFixed_transport_baseEquiv_val
      (T := K') (A := E.restrictScalars ℚ)
      (B := abstractFixedField ℚ (SeparableClosure ℚ) H') hE eK'Base c
  have hrelative :=
    rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm
      H H' hH'H hnormal
        (Additive.ofMul (ideleClassCongr eK' c))
  change
    relativeNorm rationalIdeleClassRepresentation H H' hH'H
        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
          H H' hH'H
          (Additive.ofMul (ideleClassCongr eK' c))) =
      rationalAbstractFixedFieldIdeleClassEquivFixed H
        (Additive.ofMul
          (_root_.ideleClassNorm F E
            (ideleClassCongr eK' c)))
    at hrelative
  calc
    relativeNorm rationalIdeleClassRepresentation H H' hH'H
        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
          K' L' j (Additive.ofMul c)) =
      relativeNorm rationalIdeleClassRepresentation H H' hH'H
        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
          H H' hH'H (Additive.ofMul (ideleClassCongr eK' c))) :=
      congrArg (relativeNorm rationalIdeleClassRepresentation H H' hH'H)
        hupper.symm
    _ = rationalAbstractFixedFieldIdeleClassEquivFixed H
        (Additive.ofMul
          (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) := hrelative
    _ = numberFieldEmbeddedIdeleClassEquivAmbientFixed
        K L jLower (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
      change
        rationalAbstractFixedFieldIdeleClassEquivFixed H
            (Additive.ofMul
              (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) =
          rationalAbstractFixedFieldIdeleClassEquivFixed H
            (Additive.ofMul
              (ideleClassCongr eK (_root_.ideleClassNorm K K' c)))
      apply congrArg (rationalAbstractFixedFieldIdeleClassEquivFixed H)
      apply congrArg Additive.ofMul
      exact
        (ideleClassCongr_ideleClassNorm
          (K := K) (K' := F) (L := K') (L' := E) eK eK' hcompat c).symm
theorem

globalNormResidueMonoidHomOfEmbedding_norm_restriction

L2665 ↗
GlobalClassFieldTheory.Reciprocity.globalNormResidueMonoidHomOfEmbedding_norm_restriction
Documentation

For one common compatible embedding of a Galois base-change diamond, the genuine global norm-residue maps commute with ordinary idele-class norm and actual restriction of automorphisms.

Source context 24 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
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']
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    (CyclicCohomology.extensionSubgroup
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
      (numberFieldEmbeddedBaseSubgroup K' L' j)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
noncomputable local instance
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    Finite
      ((numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
        CyclicCohomology.extensionSubgroup
          (numberFieldEmbeddedBaseSubgroup K L
            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
          (numberFieldEmbeddedBaseSubgroup K' L' j)
          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
/-- Reuse the canonical absolute fixed-field witness for the lower embedded
tower.  The base-change relative witness below needs this exact instance path
when forming the absolute finite-dimensional tower. -/
noncomputable local instance
    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedAbsoluteQuotientFinite K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsScalarTower ℚ
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    FiniteDimensional ℚ
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  FiniteDimensional.trans ℚ
    (abstractFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup K L
        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    NumberField
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  NumberField.of_module_finite ℚ
    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
noncomputable local instance
    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    IsGalois
      (abstractFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup K L
          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
  abstractRelativeFixedField_isGalois
    ℚ (SeparableClosure ℚ)
    (numberFieldEmbeddedBaseSubgroup K L
      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
    (numberFieldEmbeddedBaseSubgroup K' L' j)
    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
    (numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j)
Lean source · full declaration
theorem globalNormResidueMonoidHomOfEmbedding_norm_restriction
    [FiniteDimensional K L] [IsAbelianGalois K L]
    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
    [FiniteDimensional K K'] [IsGalois K K']
    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
      j.comp (IsScalarTower.toAlgHom ℚ L L')
    ((AlgEquiv.restrictNormalHom L).comp
        (AlgEquiv.restrictScalarsHom K)).comp
        (globalNormResidueMonoidHomOfEmbedding K' L' j) =
      (globalNormResidueMonoidHomOfEmbedding K L jLower).comp
        (_root_.ideleClassNorm K K') := by
  dsimp only
  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
    j.comp (IsScalarTower.toAlgHom ℚ L L')
  let H :=
    numberFieldEmbeddedBaseSubgroup K L jLower
  let H' :=
    numberFieldEmbeddedBaseSubgroup K' L' j
  let J :=
    numberFieldEmbeddedTopSubgroup K L jLower
  let J' :=
    numberFieldEmbeddedTopSubgroup K' L' j
  let hJH :=
    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
  let hJ'H' :=
    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
  let hH'H :=
    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
  let hJ'J :=
    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
  let _ :
      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
  let _ :
      Finite
        (H.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H J hJH) :=
    numberFieldEmbeddedExtensionQuotient_finite K L jLower
  let _ :
      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
  let _ :
      Finite
        (H'.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H' J' hJ'H') :=
    numberFieldEmbeddedExtensionQuotient_finite K' L' j
  let hHH'finite :=
    numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j
  let T :
      FiniteAbstractFieldExtension
        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
    { field := numberFieldEmbeddedFiniteAbstractField K' L' j
      base := numberFieldEmbeddedFiniteAbstractField K L jLower
      below := hH'H
      finiteQuotient := hHH'finite }
  let hTBaseNormal :
      (CyclicCohomology.extensionSubgroup
        T.base.field J hJH).Normal := by
    change
      (CyclicCohomology.extensionSubgroup H J hJH).Normal
    exact numberFieldEmbeddedExtensionSubgroup_normal K L jLower
  let hTBaseFinite :
      Finite
        (T.base.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup
            T.base.field J hJH) := by
    change
      Finite
        (H.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H J hJH)
    exact numberFieldEmbeddedExtensionQuotient_finite K L jLower
  let hTFieldNormal :
      (CyclicCohomology.extensionSubgroup
        T.field.field J' hJ'H').Normal := by
    change
      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal
    exact numberFieldEmbeddedExtensionSubgroup_normal K' L' j
  let hTFieldFinite :
      Finite
        (T.field.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup
            T.field.field J' hJ'H') := by
    change
      Finite
        (H'.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup H' J' hJ'H')
    exact numberFieldEmbeddedExtensionQuotient_finite K' L' j
  let restrictActual :
      Gal(L' / K') →* Gal(L / K) :=
    (AlgEquiv.restrictNormalHom L).comp
      (AlgEquiv.restrictScalarsHom K)
  apply MonoidHom.ext
  intro c
  let a :=
    numberFieldEmbeddedIdeleClassEquivAmbientFixed
      K' L' j (Additive.ofMul c)
  have hnat :=
    DegreeData.normResidueNaturality_norm_restriction
      (D := rationalCyclotomicDegreeData)
      (A := rationalIdeleClassRepresentation)
      (v := rationalCyclotomicIdeleClassValuationData)
      (hcf := rationalIdeleClassRepresentation_satisfiesClassFieldAxiom)
      (T := T) (L := J) (L' := J')
      (hLnormal := hTBaseNormal)
      (hL'normal := hTFieldNormal)
      (hLKfinite := hTBaseFinite)
      (hL'K'finite := hTFieldFinite)
      hJH hJ'H' hJ'J
  have hnatc :=
    DFunLike.congr_fun hnat
      (finiteNormClass rationalIdeleClassRepresentation
        H' J' hJ'H' a)
  change _ =
    rationalCyclotomicDegreeData.normResidueSymbol
      rationalIdeleClassRepresentation
      rationalCyclotomicIdeleClassValuationData
      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
      T.base
      { field := J
        below := hJH
        normal := hTBaseNormal
        finite := hTBaseFinite }
      (finiteReciprocityNaturalityNormMap
        rationalIdeleClassRepresentation
        T.base.field T.field.field J J'
        hJH hJ'H' T.below hJ'J
        (finiteNormClass rationalIdeleClassRepresentation
          T.field.field J' hJ'H' a)) at hnatc
  rw [finiteReciprocityNaturalityNormMap_finiteNormClass]
    at hnatc
  have hnorm :
      relativeNorm rationalIdeleClassRepresentation
          H H' hH'H a =
        numberFieldEmbeddedIdeleClassEquivAmbientFixed
          K L jLower
          (Additive.ofMul (_root_.ideleClassNorm K K' c)) :=
    numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
      K K' L L' j c
  calc
    restrictActual
        (globalNormResidueMonoidHomOfEmbedding K' L' j c) =
      restrictActual
        (Additive.toMul
          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
            K' L' j
            (rationalCyclotomicDegreeData.normResidueSymbol
              rationalIdeleClassRepresentation
              rationalCyclotomicIdeleClassValuationData
              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
              (numberFieldEmbeddedFiniteAbstractField K' L' j)
              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
              (finiteNormClass rationalIdeleClassRepresentation
                H' J' hJ'H' a)))) := by
      rw [globalNormResidueMonoidHomOfEmbedding_apply]
    _ =
      Additive.toMul
        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
          K L jLower
          (MonoidHom.toAdditive
            (normResidueNaturalityAbelianizedRestriction
              H H' J J' hJH hJ'H' hH'H hJ'J)
            (rationalCyclotomicDegreeData.normResidueSymbol
              rationalIdeleClassRepresentation
              rationalCyclotomicIdeleClassValuationData
              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
              (numberFieldEmbeddedFiniteAbstractField K' L' j)
              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
              (finiteNormClass rationalIdeleClassRepresentation
                H' J' hJ'H' a)))) := by
      exact
        numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
          K K' L L' j _
    _ =
      Additive.toMul
        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
          K L jLower
          (rationalCyclotomicDegreeData.normResidueSymbol
            rationalIdeleClassRepresentation
            rationalCyclotomicIdeleClassValuationData
            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
            (numberFieldEmbeddedFiniteAbstractField K L jLower)
            (numberFieldEmbeddedFiniteGaloisSubextension K L jLower)
            (finiteNormClass rationalIdeleClassRepresentation
              H J hJH
              (relativeNorm rationalIdeleClassRepresentation
                H H' hH'H a)))) := by
      exact congrArg
        (fun z =>
          Additive.toMul
            (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
              K L jLower z))
        hnatc
    _ =
      globalNormResidueMonoidHomOfEmbedding K L jLower
        (_root_.ideleClassNorm K K' c) := by
      rw [hnorm,
        ← globalNormResidueMonoidHomOfEmbedding_apply]
theorem

normResidueSymbol_restriction_sameBase

L2875 ↗
GlobalClassFieldTheory.Reciprocity.normResidueSymbol_restriction_sameBase
Documentation

Same-base norm-residue naturality in a compatible ambient: restriction from the larger finite Galois subextension commutes with the canonical projection between its finite norm quotient and the norm quotient of an intermediate subextension.

Source context 15 commands

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

open scoped IsMulCommutative
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open GlobalClassFields
open KummerTheory
open AlgebraicNumberTheory
open LocalClassFieldTheory
open RamificationTheory
universe u
attribute [local instance] naturalityIdeleClassCommGroup
local instance ideleClassGroupIsMulCommutative
    {F : Type} [Field F] [NumberField F]
    : IsMulCommutative (IdeleClassGroup F) :=
  ⟨⟨fun a b => mul_comm a b⟩⟩
local instance ideleClassSubgroupNormal
    {F : Type} [Field F] [NumberField F]
    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
  N.normal_of_isMulCommutative
variable
    {G : IntegralRepGroupType} [Group G] [TopologicalSpace G]
    [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
    [TotallyDisconnectedSpace G]
Lean source · full declaration
theorem normResidueSymbol_restriction_sameBase
    (D : DegreeData G)
    (A : Rep ℤ G)
    (v : ValuationData D A)
    (hcf : SatisfiesClassFieldAxiom A)
    (K : FiniteAbstractField G)
    (M L : ClosedSubgroup G)
    (hLM : L.toSubgroup ≤ M.toSubgroup)
    (hMK : M.toSubgroup ≤ K.field.toSubgroup)
    [hLnormal :
      (CyclicCohomology.extensionSubgroup
        K.field L (hLM.trans hMK)).Normal]
    [hMnormal :
      (CyclicCohomology.extensionSubgroup K.field M hMK).Normal]
    [hLfinite :
      Finite
        (K.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup K.field L
            (hLM.trans hMK))] :
    letI _ : Finite
        (M.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup M L hLM) :=
      abstractReciprocity_lowerExtension_finite
        K.field M L hLM hMK
    letI hIntermediateFinite : Finite
        (K.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup K.field M hMK) :=
      abstractReciprocity_intermediateQuotient_finite
        K.field M L hLM hMK
    let EM : FiniteGaloisSubextension K.field :=
      ⟨M, hMK, hMnormal, hIntermediateFinite⟩
    let EL : FiniteGaloisSubextension K.field :=
      ⟨L, hLM.trans hMK, hLnormal, hLfinite⟩
    let hEL_EM : EL.field.toSubgroup ≤ EM.field.toSubgroup :=
      hLM
    let QL : Type :=
      FiniteNormQuotient A K.field L (hLM.trans hMK)
    let QM : Type :=
      FiniteNormQuotient A K.field M hMK
    let AL : Type :=
      Additive
        (Abelianization
          (K.field.toSubgroup ⧸
            CyclicCohomology.extensionSubgroup
              K.field L (hLM.trans hMK)))
    let AM : Type :=
      Additive
        (Abelianization
          (K.field.toSubgroup ⧸
            CyclicCohomology.extensionSubgroup K.field M hMK))
    let restriction :
        AL →+ AM :=
      MonoidHom.toAdditive
        (normResidueNaturalityAbelianizedRestriction
          K.field K.field EM.field EL.field
          EM.below EL.below le_rfl hEL_EM)
    let projection :
        QL →+ QM :=
      abstractReciprocityNormProjection
        A K.field EM.field EL.field hEL_EM EM.below
    let normEL :
        QL →+ AL :=
      (DegreeData.normResidueSymbol
        (D := D) (A := A) (v := v) (hcf := hcf)
        (K := K) (L := EL)).toAddMonoidHom
    let normEM :
        QM →+ AM :=
      (DegreeData.normResidueSymbol
        (D := D) (A := A) (v := v) (hcf := hcf)
        (K := K) (L := EM)).toAddMonoidHom
    (restriction.comp normEL :
        QL →+ AM) =
      (normEM.comp projection :
        QL →+ AM) := by
  let _ : Finite
      (M.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup M L hLM) :=
    abstractReciprocity_lowerExtension_finite
      K.field M L hLM hMK
  let hIntermediateFinite : Finite
      (K.field.toSubgroup ⧸
        CyclicCohomology.extensionSubgroup K.field M hMK) :=
    abstractReciprocity_intermediateQuotient_finite
      K.field M L hLM hMK
  let EM : FiniteGaloisSubextension K.field :=
    ⟨M, hMK, hMnormal, hIntermediateFinite⟩
  let EL : FiniteGaloisSubextension K.field :=
    ⟨L, hLM.trans hMK, hLnormal, hLfinite⟩
  let hEL_EM : EL.field.toSubgroup ≤ EM.field.toSubgroup :=
    hLM
  let QL : Type :=
    FiniteNormQuotient A K.field L (hLM.trans hMK)
  let QM : Type :=
    FiniteNormQuotient A K.field M hMK
  let AL : Type :=
    Additive
      (Abelianization
        (K.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup
            K.field L (hLM.trans hMK)))
  let AM : Type :=
    Additive
      (Abelianization
        (K.field.toSubgroup ⧸
          CyclicCohomology.extensionSubgroup K.field M hMK))
  let restriction :
      AL →+ AM :=
    MonoidHom.toAdditive
      (normResidueNaturalityAbelianizedRestriction
        K.field K.field EM.field EL.field
        EM.below EL.below le_rfl hEL_EM)
  let projection :
      QL →+ QM :=
    abstractReciprocityNormProjection
      A K.field EM.field EL.field hEL_EM EM.below
  let T : FiniteAbstractFieldExtension G :=
    { base := K
      field := K
      below := le_rfl
      finiteQuotient :=
        (FiniteGaloisSubextension.refl K.field).finite }
  have hnat :=
    D.normResidueNaturality_norm_restriction
      (hLnormal := hMnormal) (hL'normal := hLnormal)
      (hLKfinite := hIntermediateFinite) (hL'K'finite := hLfinite)
      A v hcf T M L hMK (hLM.trans hMK) hLM
  rw [finiteReciprocityNaturalityNormMap_sameBase_eq_normProjection]
    at hnat
  dsimp only [T, restriction, projection, EM, EL, hEL_EM, QL, QM, AL, AM,
    FiniteGaloisSubextension.extensionQuotient] at hnat ⊢
  exact hnat

Statements and documentation are read from the Lean source at the revision shown. Source entries are not a measure of mathematical completeness.