-
Notifications
You must be signed in to change notification settings - Fork 3
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Showing
4 changed files
with
185 additions
and
121 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,132 @@ | ||
| import FltRegular.NumberTheory.Unramified | ||
| import FltRegular.NumberTheory.Hilbert92 | ||
| import FltRegular.NumberTheory.Hilbert90 | ||
| import FltRegular.NumberTheory.IdealNorm | ||
| import FltRegular.NumberTheory.RegularPrimes | ||
|
|
||
| open scoped NumberField | ||
|
|
||
| variable {K : Type*} {p : ℕ+} [hpri : Fact p.Prime] [Field K] [NumberField K] | ||
| variable [Fintype (ClassGroup (𝓞 K))] | ||
|
|
||
| attribute [-instance] instCoeOut | ||
|
|
||
| open Polynomial | ||
|
|
||
| variable {L} [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] | ||
| variable (σ : L ≃ₐ[K] L) (hσ : ∀ x, x ∈ Subgroup.zpowers σ) (hKL : FiniteDimensional.finrank K L = p) | ||
|
|
||
| variable {A B} [CommRing A] [CommRing B] [Algebra A B] [Algebra A L] [Algebra A K] | ||
| [Algebra B L] [IsScalarTower A B L] [IsScalarTower A K L] [IsFractionRing A K] | ||
| [IsIntegralClosure B A L] | ||
|
|
||
| lemma comap_span_galRestrict_eq_of_cyclic (β : B) (η : Bˣ) (hβ : η * (galRestrict A K B L σ) β = β) | ||
| (σ' : L ≃ₐ[K] L) : | ||
| (Ideal.span {β}).comap (galRestrict A K B L σ') = Ideal.span {β} := by | ||
| suffices : (Ideal.span {β}).map | ||
| (galRestrict A K B L σ'⁻¹).toRingEquiv.toRingHom = Ideal.span {β} | ||
| · rwa [RingEquiv.toRingHom_eq_coe, Ideal.map_comap_of_equiv, map_inv] at this | ||
| apply_fun (Ideal.span {·}) at hβ | ||
| rw [← Ideal.span_singleton_mul_span_singleton, Ideal.span_singleton_eq_top.mpr η.isUnit, | ||
| ← Ideal.one_eq_top, one_mul, ← Set.image_singleton, ← Ideal.map_span] at hβ | ||
| change Ideal.map (galRestrict A K B L σ : B →+* B) _ = _ at hβ | ||
| generalize σ'⁻¹ = σ' | ||
| obtain ⟨n, rfl : σ ^ n = σ'⟩ := mem_powers_iff_mem_zpowers.mpr (hσ σ') | ||
| rw [map_pow] | ||
| induction n with | ||
| | zero => | ||
| simp only [Nat.zero_eq, pow_zero, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe] | ||
| exact Ideal.map_id _ | ||
| | succ n IH => | ||
| simp only [AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toRingHom_eq_coe, pow_succ'] at IH ⊢ | ||
| conv_lhs at IH => rw [← hβ, Ideal.map_map] | ||
| exact IH | ||
|
|
||
| open FiniteDimensional in | ||
| theorem exists_not_isPrincipal_and_isPrincipal_map_aux | ||
| [IsDedekindDomain A] [IsUnramified A B] (η : Bˣ) | ||
| (hη : Algebra.norm K (algebraMap B L η) = 1) | ||
| (hη' : ¬∃ α : Bˣ, algebraMap B L η = (algebraMap B L α) / σ (algebraMap B L α)) : | ||
| ∃ I : Ideal A, ¬I.IsPrincipal ∧ (I.map (algebraMap A B)).IsPrincipal := by | ||
| obtain ⟨β, hβ_zero, hβ⟩ := Hilbert90_integral (A := A) (B := B) σ hσ η hη | ||
| haveI : IsDomain B := | ||
| (IsIntegralClosure.equiv A B L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L) | ||
| have hβ' := comap_map_eq_of_isUnramified K L _ | ||
| (comap_span_galRestrict_eq_of_cyclic σ hσ (A := A) (B := B) β η hβ) | ||
| refine ⟨(Ideal.span {β}).comap (algebraMap A B), ?_, ?_⟩ | ||
| · rintro ⟨⟨γ, hγ : _ = Ideal.span _⟩⟩ | ||
| rw [hγ, Ideal.map_span, Set.image_singleton, Ideal.span_singleton_eq_span_singleton] at hβ' | ||
| obtain ⟨a, rfl⟩ := hβ' | ||
| rw [map_mul, AlgEquiv.commutes, mul_left_comm, (mul_right_injective₀ _).eq_iff] at hβ | ||
| apply hη' | ||
| use a | ||
| conv_rhs => enter [1]; rw [← hβ] | ||
| rw [map_mul, ← AlgHom.coe_coe σ, ← algebraMap_galRestrictHom_apply A K B L σ a] | ||
| refine (mul_div_cancel _ ?_).symm | ||
| · rw [ne_eq, (injective_iff_map_eq_zero' _).mp (IsIntegralClosure.algebraMap_injective B A L), | ||
| (injective_iff_map_eq_zero' _).mp (galRestrict A K B L σ).injective] | ||
| exact a.isUnit.ne_zero | ||
| · exact (mul_ne_zero_iff.mp hβ_zero).1 | ||
| · rw [hβ'] | ||
| exact ⟨⟨_, rfl⟩⟩ | ||
|
|
||
| attribute [local instance 2000] NumberField.inst_ringOfIntegersAlgebra Algebra.toSMul Algebra.toModule | ||
|
|
||
| attribute [local instance] FractionRing.liftAlgebra | ||
|
|
||
| open FiniteDimensional (finrank) | ||
|
|
||
| theorem Ideal.isPrincipal_pow_finrank_of_isPrincipal_map [IsDedekindDomain A] (I : Ideal A) | ||
| (hI : (I.map (algebraMap A B)).IsPrincipal) : (I ^ finrank K L).IsPrincipal := by | ||
| haveI : IsDomain B := | ||
| (IsIntegralClosure.equiv A B L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L) | ||
| haveI := IsIntegralClosure.isNoetherian A K L B | ||
| haveI := IsIntegralClosure.isDedekindDomain A K L B | ||
| haveI := IsIntegralClosure.isFractionRing_of_finite_extension A K L B | ||
| have hAB : Function.Injective (algebraMap A B) | ||
| · refine Function.Injective.of_comp (f := algebraMap B L) ?_ | ||
| rw [← RingHom.coe_comp, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq A K L] | ||
| exact (algebraMap K L).injective.comp (IsFractionRing.injective _ _) | ||
| rw [← NoZeroSMulDivisors.iff_algebraMap_injective] at hAB | ||
| letI : Algebra (FractionRing A) (FractionRing B) := FractionRing.liftAlgebra _ _ | ||
| have : IsScalarTower A (FractionRing A) (FractionRing B) := | ||
| FractionRing.isScalarTower_liftAlgebra _ _ | ||
| have H : RingHom.comp (algebraMap (FractionRing A) (FractionRing B)) | ||
| ↑(FractionRing.algEquiv A K).symm.toRingEquiv = | ||
| RingHom.comp ↑(FractionRing.algEquiv B L).symm.toRingEquiv (algebraMap K L) | ||
| · apply IsLocalization.ringHom_ext (nonZeroDivisors A) | ||
| ext | ||
| simp only [AlgEquiv.toRingEquiv_eq_coe, RingHom.coe_comp, RingHom.coe_coe, | ||
| AlgEquiv.coe_ringEquiv, Function.comp_apply, AlgEquiv.commutes, | ||
| ← IsScalarTower.algebraMap_apply] | ||
| rw [IsScalarTower.algebraMap_apply A B L, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply] | ||
| have : IsSeparable (FractionRing A) (FractionRing B) := IsSeparable_of_equiv_equiv _ _ H | ||
| have hLK : finrank (FractionRing A) (FractionRing B) = finrank K L := | ||
| (FiniteDimensional.finrank_of_equiv_equiv _ _ H).symm | ||
| rw [← hLK, ← Ideal.spanIntNorm_map, ← (I.map (algebraMap A B)).span_singleton_generator, | ||
| Ideal.spanIntNorm_singleton] | ||
| exact ⟨⟨_, rfl⟩⟩ | ||
|
|
||
| theorem exists_not_isPrincipal_and_isPrincipal_map (K L : Type*) | ||
| [Field K] [Field L] [NumberField K] [NumberField L] [Algebra K L] | ||
| [FiniteDimensional K L] [IsGalois K L] [IsUnramified ↥(𝓞 K) ↥(𝓞 L)] [IsCyclic (L ≃ₐ[K] L)] | ||
| [NumberField.InfinitePlace.IsUnramified K L] (hKL : Nat.Prime (finrank K L)) : | ||
| ∃ I : Ideal (𝓞 K), ¬I.IsPrincipal ∧ (I.map (algebraMap ↥(𝓞 K) ↥(𝓞 L))).IsPrincipal := by | ||
| obtain ⟨⟨σ, hσ⟩⟩ := ‹IsCyclic (L ≃ₐ[K] L)› | ||
| obtain ⟨η, hη, hη'⟩ := Hilbert92 hKL σ hσ | ||
| exact exists_not_isPrincipal_and_isPrincipal_map_aux (A := ↥(𝓞 K)) σ hσ η hη (not_exists.mpr hη') | ||
|
|
||
| /-- This is **Hilbert Theorem 94**, which states that if `L/K` is an unramified | ||
| cyclic finite extension of number fields of prime degree, | ||
| then the degree divides the class number of `K`. -/ | ||
| theorem dvd_card_classGroup_of_isUnramified_isCyclic (K L : Type*) | ||
| [Field K] [Field L] [NumberField K] [NumberField L] [Algebra K L] | ||
| [FiniteDimensional K L] [IsGalois K L] [IsUnramified ↥(𝓞 K) ↥(𝓞 L)] [IsCyclic (L ≃ₐ[K] L)] | ||
| [NumberField.InfinitePlace.IsUnramified K L] (hKL : Nat.Prime (finrank K L)) : | ||
| finrank K L ∣ Fintype.card (ClassGroup ↥(𝓞 K)) := by | ||
| obtain ⟨I, hI, hI'⟩ := exists_not_isPrincipal_and_isPrincipal_map K L hKL | ||
| letI := Fact.mk hKL | ||
| rw [← Int.ofNat_dvd, (Nat.prime_iff_prime_int.mp hKL).irreducible.dvd_iff_not_coprime, | ||
| Nat.isCoprime_iff_coprime] | ||
| exact fun h ↦ hI (IsPrincipal_of_IsPrincipal_pow_of_Coprime _ _ h _ | ||
| (Ideal.isPrincipal_pow_finrank_of_isPrincipal_map _ hI')) |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Oops, something went wrong.