Module

IdeleClassDirectLimitFiniteTowerNormProof.lean

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.IdeleClassDirectLimitFiniteTowerNormProof

60 declarations indexed from sourceView source ↗

Module documentation

Proof of ordinary norm comparison in finite towers of rational fixed fields

This proof leaf installs the canonical finite-tower context once and splits the normal-closure embedding calculation, the pointwise coset action, and the finite product calculation into separate commands. The public theorem is a thin wrapper around those providers.

Imports1

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.IdeleClassDirectLimitFiniteTowerNormStatement

Imported by2

abbrev

rationalFiniteTowerRestrictedNormal

L44 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRestrictedNormal

Source context 12 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)

Lean source · statement

abbrev rationalFiniteTowerRestrictedNormal :
    IntermediateField ℚ (SeparableClosure ℚ)

abbrev

rationalFiniteTowerRestrictedUpper

L47 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRestrictedUpper

Source context 12 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)

Lean source · statement

abbrev rationalFiniteTowerRestrictedUpper :
    IntermediateField ℚ (SeparableClosure ℚ)

instance

rationalFiniteTowerLowerFiniteDimensional

L58 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerFiniteDimensional

Source context 16 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK

Lean source · statement

instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀

instance

rationalFiniteTowerUpperFiniteDimensional

L63 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerUpperFiniteDimensional

Source context 17 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite

Lean source · statement

instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀

instance

rationalFiniteTowerLowerUpperScalarTower

L68 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerUpperScalarTower

Source context 18 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite

Lean source · statement

instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀

instance

rationalFiniteTowerUpperAbsoluteFiniteDimensional

L72 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerUpperAbsoluteFiniteDimensional

Source context 19 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀

instance

rationalFiniteTowerRestrictedUpperFiniteDimensional

L76 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRestrictedUpperFiniteDimensional

Source context 20 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀

Lean source · statement

instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀

instance

rationalFiniteTowerLowerNumberField

L81 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerNumberField

Source context 21 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance

Lean source · statement

instance rationalFiniteTowerLowerNumberField :
    NumberField F₀

instance

rationalFiniteTowerUpperNumberField

L85 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerUpperNumberField

Source context 22 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀

Lean source · statement

instance rationalFiniteTowerUpperNumberField :
    NumberField E₀

instance

rationalFiniteTowerUpperSeparableClosureScalarTower

L89 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerUpperSeparableClosureScalarTower

Source context 23 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀

Lean source · statement

instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ)

theorem

rationalFiniteTower_lower_le_upper

L96 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_lower_le_upper

Source context 25 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
omit hKfinite hfinite in

Lean source · statement

theorem rationalFiniteTower_lower_le_upper :
    F₀ ≤ IntermediateField.restrictScalars ℚ E₀

theorem

rationalFiniteTower_upper_le_normal

L101 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_upper_le_normal

Source context 25 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
omit hKfinite hfinite in

Lean source · statement

theorem rationalFiniteTower_upper_le_normal :
    IntermediateField.restrictScalars ℚ E₀ ≤ NQ₀

theorem

rationalFiniteTower_lower_le_normal

L107 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_lower_le_normal

Source context 25 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
omit hKfinite hfinite in

Lean source · statement

theorem rationalFiniteTower_lower_le_normal :
    F₀ ≤ NQ₀

instance

rationalFiniteTowerNormalScalarTower

L112 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalScalarTower

Source context 24 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)

Lean source · statement

instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀

instance

rationalFiniteTowerNormalFiniteDimensional

L116 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalFiniteDimensional

Source context 25 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀

instance

rationalFiniteTowerNormalAbsoluteFiniteDimensional

L120 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalAbsoluteFiniteDimensional

Source context 26 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)

Lean source · statement

instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀

instance

rationalFiniteTowerNormalNumberField

L124 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalNumberField

Source context 27 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀

Lean source · statement

instance rationalFiniteTowerNormalNumberField :
    NumberField N₀

instance

rationalFiniteTowerRestrictedNormalFiniteDimensional

L128 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRestrictedNormalFiniteDimensional

Source context 28 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀

Lean source · statement

instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀

theorem

rationalFiniteTower_normal_le_rationalNormalClosure

L133 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_normal_le_rationalNormalClosure

Source context 29 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance

Lean source · statement

theorem rationalFiniteTower_normal_le_rationalNormalClosure :
    NQ₀ ≤ (U₀ : IntermediateField ℚ (SeparableClosure ℚ))

theorem

rationalFiniteTower_lower_le_rationalNormalClosure

L139 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_lower_le_rationalNormalClosure

Source context 29 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance

Lean source · statement

theorem rationalFiniteTower_lower_le_rationalNormalClosure :
    F₀ ≤ (U₀ : IntermediateField ℚ (SeparableClosure ℚ))

theorem

rationalFiniteTower_upper_le_rationalNormalClosure

L144 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_upper_le_rationalNormalClosure

Source context 29 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance

Lean source · statement

theorem rationalFiniteTower_upper_le_rationalNormalClosure :
    IntermediateField.restrictScalars ℚ E₀ ≤
      (U₀ : IntermediateField ℚ (SeparableClosure ℚ))

instance

rationalFiniteTowerRationalNormalClosureLowerAlgebra

L150 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureLowerAlgebra

Source context 29 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀

instance

rationalFiniteTowerRationalNormalClosureUpperAlgebra

L155 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureUpperAlgebra

Source context 30 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀

instance

rationalFiniteTowerRationalNormalClosureNormalAlgebra

L160 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureNormalAlgebra

Source context 31 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀

instance

rationalFiniteTowerRationalNormalClosureLowerScalarTower

L165 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureLowerScalarTower

Source context 32 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀

instance

rationalFiniteTowerRationalNormalClosureUpperScalarTower

L169 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureUpperScalarTower

Source context 33 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀

instance

rationalFiniteTowerRationalNormalClosureNormalScalarTower

L173 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalNormalClosureNormalScalarTower

Source context 34 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀

instance

rationalFiniteTowerLowerUpperClosureScalarTower

L177 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerUpperClosureScalarTower

Source context 35 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)

Lean source · statement

instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀

instance

rationalFiniteTowerLowerNormalClosureScalarTower

L183 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerNormalClosureScalarTower

Source context 36 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)

Lean source · statement

instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀

instance

rationalFiniteTowerClosureSeparableClosureScalarTower

L189 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureSeparableClosureScalarTower

Source context 37 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)

Lean source · statement

instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ)

instance

rationalFiniteTowerClosureOverUpperFiniteDimensional

L195 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureOverUpperFiniteDimensional

Source context 38 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)

Lean source · statement

instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀

instance

rationalFiniteTowerClosureOverLowerFiniteDimensional

L199 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureOverLowerFiniteDimensional

Source context 39 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀

Lean source · statement

instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀

instance

rationalFiniteTowerClosureOverNormalFiniteDimensional

L203 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureOverNormalFiniteDimensional

Source context 40 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀

Lean source · statement

instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀

instance

rationalFiniteTowerLowerSeparableClosureGalois

L207 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerSeparableClosureGalois

Source context 41 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀

Lean source · statement

instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ)

instance

rationalFiniteTowerNormalGalois

L211 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalGalois

Source context 42 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)

Lean source · statement

instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀

instance

rationalFiniteTowerClosureOverLowerGalois

L214 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureOverLowerGalois

Source context 43 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)

Lean source · statement

instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀

instance

rationalFiniteTowerClosureOverNormalGalois

L217 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerClosureOverNormalGalois

Source context 44 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀

Lean source · statement

instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀

instance

rationalFiniteTowerQuotientFintype

L220 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerQuotientFintype

Source context 45 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀

Lean source · statement

instance rationalFiniteTowerQuotientFintype : Fintype Q₀

def

rationalFiniteTowerCosetEquiv

L223 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerCosetEquiv

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerCosetEquiv :
    Q₀ ≃ (E₀ →ₐ[F₀] N₀)

def

rationalFiniteTowerNormalInclusion

L230 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalInclusion

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerNormalInclusion :
    N₀ →ₐ[F₀] U₀

def

rationalFiniteTowerUpperInclusion

L237 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerUpperInclusion

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerUpperInclusion :
    E₀ →ₐ[ℚ] U₀

def

rationalFiniteTowerRelativeClass

L242 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRelativeClass

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerRelativeClass
    (c : IdeleClassGroup E₀) :
    RelativeIdeleGroup.ClassGroup F₀ E₀

def

rationalFiniteTowerRationalRelativeClass

L248 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerRationalRelativeClass

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerRationalRelativeClass
    (c : IdeleClassGroup E₀) :
    RelativeIdeleGroup.ClassGroup ℚ E₀

def

rationalFiniteTowerEmbeddedClass

L254 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerEmbeddedClass

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

def rationalFiniteTowerEmbeddedClass
    (c : IdeleClassGroup E₀) (q : Q₀) : IdeleClassGroup U₀

theorem

rationalFiniteTowerCoset_comp_normalInclusion

L263 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerCoset_comp_normalInclusion

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTowerCoset_comp_normalInclusion
    (q : Q₀) :
    (IsScalarTower.toAlgHom F₀ U₀ (SeparableClosure ℚ)).comp
        ((rationalFiniteTowerNormalInclusion K L hLK).comp
          (rationalFiniteTowerCosetEquiv K L hLK q)) =
      abstractFixedFieldCosetToAlgHom
        ℚ (SeparableClosure ℚ) K L hLK q

theorem

rationalFiniteTower_restrictedEmbedding

L285 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_restrictedEmbedding

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_restrictedEmbedding
    (sigma : K.toSubgroup) :
    (AlgEquiv.restrictNormalHom U₀ sigma.1).toAlgHom.comp
        (rationalFiniteTowerUpperInclusion K L hLK) =
      ((rationalFiniteTowerNormalInclusion K L hLK).comp
        (rationalFiniteTowerCosetEquiv K L hLK
          (QuotientGroup.mk sigma))).restrictScalars ℚ

theorem

rationalFiniteTower_representativeAction

L325 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_representativeAction

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_representativeAction
    (c : IdeleClassGroup E₀) (sigma : K.toSubgroup) :
    Additive.toMul
        (Additive.ofMul
          (sigma.1 •
            rationalIntermediateIdeleClassToDirectLimit EQ₀ c)) =
      rationalIntermediateIdeleClassToDirectLimit U₀
        (_root_.relativeIdeleClassBaseChangeMulEquiv
          (K := ℚ) (L := U₀)
          (RelativeIdeleGroup.classEmbedding
            ((AlgEquiv.restrictNormalHom U₀ sigma.1).toAlgHom.comp
              (rationalFiniteTowerUpperInclusion K L hLK))
            (rationalFiniteTowerRationalRelativeClass K L hLK c)))

theorem

rationalRelativeFixedFieldCosetActionPointwise

L361 ↗

GlobalClassFieldTheory.Reciprocity.rationalRelativeFixedFieldCosetActionPointwise

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalRelativeFixedFieldCosetActionPointwise
    (c : IdeleClassGroup E₀) (q : Q₀) :
    Additive.toMul
        ((relativeCosetAction rationalIdeleClassRepresentation
          K L hLK
          (rationalAbstractRelativeFixedFieldIdeleClassEquivFixedOfFiniteDimensional
            K L hLK (Additive.ofMul c)) q :
          Additive rationalIdeleClassDirectLimit)) =
      rationalIntermediateIdeleClassToDirectLimit
        (U₀ : IntermediateField ℚ (SeparableClosure ℚ))
        (rationalFiniteTowerEmbeddedClass K L hLK c q)

theorem

rationalFiniteTower_product_embeddedClass

L417 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_product_embeddedClass

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_product_embeddedClass
    (c : IdeleClassGroup E₀) :
    ∏ q : Q₀, rationalFiniteTowerEmbeddedClass K L hLK c q =
      _root_.relativeIdeleClassBaseChangeMulEquiv
        (K := F₀) (L := U₀)
        (RelativeIdeleGroup.classEmbedding
          (rationalFiniteTowerNormalInclusion K L hLK)
          (RelativeIdeleGroup.classInclusion F₀ N₀
            (RelativeIdeleGroup.classNorm F₀ E₀
              (rationalFiniteTowerRelativeClass K L hLK c))))

theorem

rationalFiniteTowerNormalInclusion_eq_toAlgHom

L433 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerNormalInclusion_eq_toAlgHom

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTowerNormalInclusion_eq_toAlgHom :
    rationalFiniteTowerNormalInclusion K L hLK =
      IsScalarTower.toAlgHom F₀ N₀ U₀

theorem

rationalFiniteTowerLowerInclusion_eq_toAlgHom

L440 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTowerLowerInclusion_eq_toAlgHom

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTowerLowerInclusion_eq_toAlgHom :
    IntermediateField.inclusion
        (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK) =
      IsScalarTower.toAlgHom ℚ F₀ U₀

theorem

rationalFiniteTower_ideleClassExtension_comp_apply

L448 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_ideleClassExtension_comp_apply

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_ideleClassExtension_comp_apply
    (x : IdeleClassGroup F₀) :
    _root_.ideleClassExtension N₀ U₀
        (_root_.ideleClassExtension F₀ N₀ x) =
      _root_.ideleClassExtension F₀ U₀ x

theorem

rationalFiniteTower_baseChange_classInclusion

L461 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_baseChange_classInclusion

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_baseChange_classInclusion
    (x : IdeleClassGroup F₀) :
    _root_.relativeIdeleClassBaseChangeMulEquiv
        (K := F₀) (L := N₀)
        (RelativeIdeleGroup.classInclusion F₀ N₀ x) =
      _root_.ideleClassExtension F₀ N₀ x

theorem

rationalFiniteTower_baseChange_normalInclusion_classInclusion

L470 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_baseChange_normalInclusion_classInclusion

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem
    rationalFiniteTower_baseChange_normalInclusion_classInclusion
    (x : IdeleClassGroup F₀) :
    _root_.relativeIdeleClassBaseChangeMulEquiv
        (K := F₀) (L := U₀)
        (RelativeIdeleGroup.classEmbedding
          (rationalFiniteTowerNormalInclusion K L hLK)
          (RelativeIdeleGroup.classInclusion F₀ N₀ x)) =
      _root_.ideleClassExtension F₀ U₀ x

theorem

rationalFiniteTower_directLimit_ideleClassExtension

L487 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_directLimit_ideleClassExtension

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_directLimit_ideleClassExtension
    (x : IdeleClassGroup F₀) :
    rationalIntermediateIdeleClassToDirectLimit
        (U₀ : IntermediateField ℚ (SeparableClosure ℚ))
        (_root_.ideleClassExtension F₀ U₀ x) =
      rationalIntermediateIdeleClassToDirectLimit F₀ x

theorem

rationalFiniteTower_relativeClassNorm_eq_ordinaryNorm

L505 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_relativeClassNorm_eq_ordinaryNorm

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_relativeClassNorm_eq_ordinaryNorm
    (c : IdeleClassGroup E₀) :
    RelativeIdeleGroup.classNorm F₀ E₀
        (rationalFiniteTowerRelativeClass K L hLK c) =
      _root_.ideleClassNorm F₀ E₀ c

theorem

rationalFiniteTower_directLimit_extension_eq_norm

L526 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_directLimit_extension_eq_norm

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_directLimit_extension_eq_norm
    (c : IdeleClassGroup E₀) :
    rationalIntermediateIdeleClassToDirectLimit
        (U₀ : IntermediateField ℚ (SeparableClosure ℚ))
        (_root_.relativeIdeleClassBaseChangeMulEquiv
          (K := F₀) (L := U₀)
          (RelativeIdeleGroup.classEmbedding
            (rationalFiniteTowerNormalInclusion K L hLK)
            (RelativeIdeleGroup.classInclusion F₀ N₀
              (RelativeIdeleGroup.classNorm F₀ E₀
                (rationalFiniteTowerRelativeClass K L hLK c))))) =
      rationalIntermediateIdeleClassToDirectLimit F₀
        (_root_.ideleClassNorm F₀ E₀ c)

theorem

rationalFiniteTower_product_embeddings_eq_norm

L570 ↗

GlobalClassFieldTheory.Reciprocity.rationalFiniteTower_product_embeddings_eq_norm

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalFiniteTower_product_embeddings_eq_norm
    (c : IdeleClassGroup E₀) :
    ∏ q : Q₀,
        rationalIntermediateIdeleClassToDirectLimit
          (U₀ : IntermediateField ℚ (SeparableClosure ℚ))
          (rationalFiniteTowerEmbeddedClass K L hLK c q) =
      rationalIntermediateIdeleClassToDirectLimit F₀
        (_root_.ideleClassNorm F₀ E₀ c)

theorem

rationalAbstractRelativeFixedFieldNormCore

L602 ↗

GlobalClassFieldTheory.Reciprocity.rationalAbstractRelativeFixedFieldNormCore

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem rationalAbstractRelativeFixedFieldNormCore
    (c : IdeleClassGroup E₀) :
    relativeNorm rationalIdeleClassRepresentation K L hLK
        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixedOfFiniteDimensional
          K L hLK (Additive.ofMul c)) =
      rationalAbstractFixedFieldIdeleClassEquivFixed K
        (Additive.ofMul (_root_.ideleClassNorm F₀ E₀ c))

theorem

rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm_ofFiniteTower

L638 ↗

GlobalClassFieldTheory.Reciprocity.rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm_ofFiniteTower

Documentation

The abstract relative norm agrees with the ordinary idèle-class norm for every finite tower of rational fixed fields.

Source context 46 commands

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

noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity
open ClassFormation
open LocalClassFieldTheory
open CyclicCohomology
open FiniteTowerNormCore
section RationalFiniteTower
variable
    (K L : ClosedSubgroup
      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
    (hLK : L.toSubgroup ≤ K.toSubgroup)
    [hKfinite : Finite (rationalFixedFieldAbsoluteQuotient K)]
    [hfinite : Finite
      (rationalFixedFieldRelativeQuotient K L hLK)]
local notation "F₀" =>
  abstractFixedField ℚ (SeparableClosure ℚ) K
local notation "E₀" =>
  abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
local notation "N₀" =>
  IntermediateField.normalClosure F₀ E₀ (SeparableClosure ℚ)
local notation "EQ₀" =>
  rationalFiniteTowerRestrictedUpper K L hLK
local notation "NQ₀" =>
  rationalFiniteTowerRestrictedNormal K L hLK
local notation "U₀" => rationalNormalClosure NQ₀
local notation "Q₀" =>
  K.toSubgroup ⧸ extensionSubgroup K L hLK
noncomputable local instance rationalFiniteTowerLowerFiniteDimensional :
    FiniteDimensional ℚ F₀ :=
  abstractFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K hKfinite
noncomputable local instance rationalFiniteTowerUpperFiniteDimensional :
    FiniteDimensional F₀ E₀ :=
  abstractRelativeFixedField_finiteDimensional
    ℚ (SeparableClosure ℚ) K L hLK hKfinite hfinite
local instance rationalFiniteTowerLowerUpperScalarTower :
    IsScalarTower ℚ F₀ E₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerUpperAbsoluteFiniteDimensional :
    FiniteDimensional ℚ E₀ :=
  FiniteDimensional.trans ℚ F₀ E₀
noncomputable local instance rationalFiniteTowerRestrictedUpperFiniteDimensional :
    FiniteDimensional ℚ EQ₀ := by
  change FiniteDimensional ℚ E₀
  infer_instance
noncomputable local instance rationalFiniteTowerLowerNumberField :
    NumberField F₀ :=
  NumberField.of_module_finite ℚ F₀
noncomputable local instance rationalFiniteTowerUpperNumberField :
    NumberField E₀ :=
  NumberField.of_module_finite ℚ E₀
local instance rationalFiniteTowerUpperSeparableClosureScalarTower :
    IsScalarTower F₀ E₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerNormalScalarTower :
    IsScalarTower ℚ F₀ N₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
noncomputable local instance rationalFiniteTowerNormalFiniteDimensional :
    FiniteDimensional F₀ N₀ :=
  normalClosure.is_finiteDimensional F₀ E₀ (SeparableClosure ℚ)
noncomputable local instance rationalFiniteTowerNormalAbsoluteFiniteDimensional :
    FiniteDimensional ℚ N₀ :=
  FiniteDimensional.trans ℚ F₀ N₀
noncomputable local instance rationalFiniteTowerNormalNumberField :
    NumberField N₀ :=
  NumberField.of_module_finite ℚ N₀
noncomputable local instance rationalFiniteTowerRestrictedNormalFiniteDimensional :
    FiniteDimensional ℚ NQ₀ := by
  change FiniteDimensional ℚ N₀
  infer_instance
local instance rationalFiniteTowerRationalNormalClosureLowerAlgebra :
    Algebra F₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_lower_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureUpperAlgebra :
    Algebra E₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_upper_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureNormalAlgebra :
    Algebra N₀ U₀ :=
  (IntermediateField.inclusion
    (rationalFiniteTower_normal_le_rationalNormalClosure K L hLK)).toRingHom.toAlgebra
local instance rationalFiniteTowerRationalNormalClosureLowerScalarTower :
    IsScalarTower ℚ F₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureUpperScalarTower :
    IsScalarTower ℚ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerRationalNormalClosureNormalScalarTower :
    IsScalarTower ℚ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
local instance rationalFiniteTowerLowerUpperClosureScalarTower :
    IsScalarTower F₀ E₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerLowerNormalClosureScalarTower :
    IsScalarTower F₀ N₀ U₀ :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
local instance rationalFiniteTowerClosureSeparableClosureScalarTower :
    IsScalarTower F₀ U₀ (SeparableClosure ℚ) :=
  IsScalarTower.of_algebraMap_eq' (by
    ext x
    rfl)
noncomputable local instance rationalFiniteTowerClosureOverUpperFiniteDimensional :
    FiniteDimensional E₀ U₀ :=
  FiniteDimensional.right ℚ E₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverLowerFiniteDimensional :
    FiniteDimensional F₀ U₀ :=
  FiniteDimensional.right ℚ F₀ U₀
noncomputable local instance rationalFiniteTowerClosureOverNormalFiniteDimensional :
    FiniteDimensional N₀ U₀ :=
  FiniteDimensional.right ℚ N₀ U₀
local instance rationalFiniteTowerLowerSeparableClosureGalois :
    IsGalois F₀ (SeparableClosure ℚ) :=
  IsGalois.tower_top_of_isGalois ℚ F₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerNormalGalois : IsGalois F₀ N₀ :=
  IsGalois.normalClosure F₀ E₀ (SeparableClosure ℚ)
local instance rationalFiniteTowerClosureOverLowerGalois : IsGalois F₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ F₀ U₀
local instance rationalFiniteTowerClosureOverNormalGalois : IsGalois N₀ U₀ :=
  IsGalois.tower_top_of_isGalois ℚ N₀ U₀
noncomputable local instance rationalFiniteTowerQuotientFintype : Fintype Q₀ :=
  Fintype.ofFinite Q₀

Lean source · statement

theorem
    rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm_ofFiniteTower :
    rationalAbstractRelativeFixedFieldNormStatement K L hLK

Module documentation

Proof of ordinary norm comparison in finite towers of rational fixed fields

Declarations