NYLL
Module

Exists_constantFieldDescent_finset.lean

Nyll.AlgebraicCurves.FunctionFields.Exists_constantFieldDescent_finset
7 declarations indexed from sourceView source ↗

Module documentation

Exists constant Field Descent finset

Imports8
Imported by1

Declarations

theorem

nonempty_ringHom_complex

L68 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.nonempty_ringHom_complex
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem nonempty_ringHom_complex (k : Type u) [Field k] [CharZero k] [Countable k] :
    Nonempty (k →+* ℂ)
theorem

isAlgebraic_of_rational_relation

L94 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.isAlgebraic_of_rational_relation
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem isAlgebraic_of_rational_relation {k Φ : Type*} [Field k] [Field Φ] [Algebra k Φ]
    (x y : Φ) (J : Finset ℕ) (r s : ℕ → k[X])
    (hrel : ∑ j ∈ J, (aeval x (r j) / aeval x (s j)) * y ^ j = 0)
    (hne : ∃ j ∈ J, aeval x (r j) / aeval x (s j) ≠ 0) :
    IsAlgebraic k⟮x⟯ y
theorem

descent_abstract

L123 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.descent_abstract
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem descent_abstract {k F : Type*} [Field k] [Field F] [Algebra k F]
    {x y : F} (hx : Transcendental k x) (J : Finset ℕ) (r s : ℕ → k[X])
    (hrel : ∑ j ∈ J, (aeval x (r j) / aeval x (s j)) * y ^ j = 0)
    (hne : ∃ j ∈ J, aeval x (r j) / aeval x (s j) ≠ 0) :
    ∃ x₀ : k⟮x, y⟯, (x₀ : F) = x ∧ Transcendental k x₀ ∧
      FiniteDimensional k⟮x₀⟯ k⟮x, y⟯
theorem

aeval_mem_of_lift

L168 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.aeval_mem_of_lift
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem aeval_mem_of_lift {K₀ K F : Type*} [Field K₀] [Field K] [Field F] [Algebra K₀ K]
    [Algebra K F] [Algebra K₀ F] [IsScalarTower K₀ K F]
    {M : IntermediateField K₀ F} {x : F} (hx : x ∈ M) {p : K[X]}
    (hp : ∃ p₀ : K₀[X], p₀.map (algebraMap K₀ K) = p) : aeval x p ∈ M
theorem

aeval_mem_of_repr

L179 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.aeval_mem_of_repr
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem aeval_mem_of_repr {K₀ K F : Type*} [Field K₀] [Field K] [Field F] [Algebra K₀ K]
    [Algebra K F] [Algebra K₀ F] [IsScalarTower K₀ K F]
    {M : IntermediateField K₀ F} {x y : F} (hx : x ∈ M) (hy : y ∈ M)
    (P : K⟮x⟯[X]) (rp sp : ℕ → K[X])
    (hrep : ∀ i, ((P.coeff i : K⟮x⟯) : F) = aeval x (rp i) / aeval x (sp i))
    (hr : ∀ i ∈ P.support, ∃ r₀ : K₀[X], r₀.map (algebraMap K₀ K) = rp i)
    (hs : ∀ i ∈ P.support, ∃ s₀ : K₀[X], s₀.map (algebraMap K₀ K) = sp i) :
    aeval y P ∈ M
theorem

core

L193 ↗
AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset.W7XDesc.core
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
namespace W7XDesc
Lean source · statement
theorem core (K : Type u) (F : Type v) [Field K] [Field F]
    [Algebra K F] [IsAlgClosed K] [CharZero K]
    {x : F} (hx : Transcendental K x) (hfin : FiniteDimensional K⟮x⟯ F) (t : Finset F) :
    ∃ (K₀ : Type u) (F₀ : Type v) (_ : Field K₀) (_ : Field F₀)
      (_ : Algebra K₀ F₀) (_ : Algebra K₀ K) (_ : Algebra F₀ F) (_ : Algebra K₀ F)
      (_ : IsScalarTower K₀ K F) (_ : IsScalarTower K₀ F₀ F) (_ : Algebra K₀ ℂ)
      (_ : IsAlgClosed K₀) (_ : Countable K₀) (_ : CharZero K₀),
      (∃ x₀ : F₀, Transcendental K₀ x₀ ∧ FiniteDimensional K₀⟮x₀⟯ F₀) ∧
      IntermediateField.adjoin K (Set.range (algebraMap F₀ F)) = ⊤ ∧
      (↑t : Set F) ⊆ Set.range (algebraMap F₀ F)
theorem

exists_constantFieldDescent_finset

L324 ↗
AlgebraicCurve.exists_constantFieldDescent_finset
Source context 8 commands

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

section ProofImplementation
namespace AlgebraicCurves.FunctionFields.Proofs.Exists_constantFieldDescent_finset
attribute [-instance]
  AlgebraicCurve.instFundamentalIdentityOfSumRamificationInertia
  instIsScalarTowerResidueFieldRestrictPushforward
  instAlgebraResidueFieldRestrictPushforward
  instIsLocalHomRestrictInclusion
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
attribute [-simp] AlgebraicCurve.Divisor.mapRestrict_single
  AlgebraicCurve.Divisor.pushforward_single AlgebraicCurve.Place.coe_restrictInclusion
  AlgebraicCurve.Place.mem_fiber AlgebraicCurve.Place.restrict_toValuationSubring
  AlgebraicCurve.Divisor.degree_pushforward AlgebraicCurve.Place.restrictResidueMap_residue
  AlgebraicCurve.Pic0.coe_pushforwardDegZeroHom AlgebraicCurve.Pic0.coe_pullbackDegZeroHom
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
  AlgebraicCurve.Divisor.evalFun_zero AlgebraicCurve.Place.evalAt_one
open Polynomial IntermediateField Cardinal
universe u v
open AlgebraicCurve
Lean source · statement
theorem _root_.AlgebraicCurve.exists_constantFieldDescent_finset (K : Type u) (F : Type v)
    [Field K] [Field F] [Algebra K F] [IsAlgClosed K] [CharZero K]
    (hfg : ∃ x : F, Transcendental K x ∧ FiniteDimensional (IntermediateField.adjoin K ({x} : Set
      F)) F)
    [IsCurveOver K F] (t : Finset F) :
    ∃ (K₀ : Type u) (F₀ : Type v) (_ : Field K₀) (_ : Field F₀)
      (_ : Algebra K₀ F₀) (_ : Algebra K₀ K) (_ : Algebra F₀ F) (_ : Algebra K₀ F)
      (_ : IsScalarTower K₀ K F) (_ : IsScalarTower K₀ F₀ F) (_ : Algebra K₀ ℂ)
      (_ : IsAlgClosed K₀) (_ : Countable K₀) (_ : IsCurveOver K₀ F₀),
      (∃ x : F₀, Transcendental K₀ x ∧
        FiniteDimensional (IntermediateField.adjoin K₀ ({x} : Set F₀)) F₀) ∧
      IntermediateField.adjoin K (Set.range (algebraMap F₀ F)) = ⊤ ∧
      (↑t : Set F) ⊆ Set.range (algebraMap F₀ F)

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