| {"id":"lemma:alt_definition_of_norm","label":"alt_definition_of_norm","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(\\alpha \\in K\\) and let \\(\\sigma _i\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Then \\[ N_{K/\\mathbb {Q}}(\\alpha )=\\prod _i \\sigma _i(\\alpha ) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.norm_eq_prod_embeddings","latex_link":"sect0002.html#lemma:alt_definition_of_norm","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.norm_eq_prod_embeddings","lean_decl_name":"Algebra.norm_eq_prod_embeddings","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Norm/Basic.lean","lean4_code":"theorem norm_eq_prod_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsAlgClosed E]\n (x : L) : algebraMap K E (norm K x) = ∏ σ : L →ₐ[K] E, σ x := by\n have hx := Algebra.IsSeparable.isIntegral K x\n rw [norm_eq_norm_adjoin K x, RingHom.map_pow, ← adjoin.powerBasis_gen hx,\n norm_eq_prod_embeddings_gen E (adjoin.powerBasis hx) (IsAlgClosed.splits_codomain _)]\n · exact (prod_embeddings_eq_finrank_pow L (L := K⟮x⟯) E (adjoin.powerBasis hx)).symm\n · haveI := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L\n exact Algebra.IsSeparable.isSeparable K _\n\n","line":219,"end_line":227,"proof_status":"formalized"} |
| {"id":"lem:lin_indep_iff_disc_ne_zero","label":"lin_indep_iff_disc_ne_zero","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_norm","defn_of_disc"],"nlp_text":"Let \\(L/K\\) be an extension of fields and let \\(B=\\{ b_1,\\dots ,b_n\\} \\) be a \\(K\\)-basis of \\(L\\). Then \\(\\Delta (B) \\neq 0\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_not_zero_of_basis","latex_link":"sect0002.html#lem:lin_indep_iff_disc_ne_zero","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_not_zero_of_basis","lean_decl_name":"Algebra.discr_not_zero_of_basis","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_not_zero_of_basis [Algebra.IsSeparable K L] (b : Basis ι K L) :\n discr K b ≠ 0 := by\n rw [discr_def, traceMatrix_of_basis, ← LinearMap.BilinForm.nondegenerate_iff_det_ne_zero]\n exact traceForm_nondegenerate _ _\n\n/-- If `b` is a basis of a finite separable field extension `L/K`,\n then `Algebra.discr K b` is a unit. -/\n","line":127,"end_line":133,"proof_status":"formalized"} |
| {"id":"lemma:num_field_disc_in_terms_of_norm","label":"num_field_disc_in_terms_of_norm","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_norm","lemma:disc_of_prim_elt_basis","lemma:diff_of_irr_pol"],"nlp_text":"Let \\(K=\\mathbb {Q}(\\alpha )\\) be a number field with \\(n=[K:\\mathbb {Q}]\\) and let \\(B=\\{ 1,\\alpha ,\\alpha ^2,\\dots ,\\alpha ^{n-1}\\} \\). Then \\[ \\Delta (B)=(-1)^{\\frac{n(n-1)}{2}}N_{K/\\mathbb {Q}}(m_\\alpha '(\\alpha )) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_norm","latex_link":"sect0002.html#lemma:num_field_disc_in_terms_of_norm","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_norm","lean_decl_name":"Algebra.discr_powerBasis_eq_norm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_powerBasis_eq_norm [Algebra.IsSeparable K L] :\n discr K pb.basis =\n (-1) ^ (n * (n - 1) / 2) *\n norm K (aeval pb.gen (derivative (R := K) (minpoly K pb.gen))) := by\n let E := AlgebraicClosure L\n letI := fun a b : E => Classical.propDecidable (Eq a b)\n have e : Fin pb.dim ≃ (L →ₐ[K] E) := by\n refine equivOfCardEq ?_\n rw [Fintype.card_fin, AlgHom.card]\n exact (PowerBasis.finrank pb).symm\n have hnodup : ((minpoly K pb.gen).aroots E).Nodup :=\n nodup_roots (Separable.map (Algebra.IsSeparable.isSeparable K pb.gen))\n have hroots : ∀ σ : L →ₐ[K] E, σ pb.gen ∈ (minpoly K pb.gen).aroots E := by\n intro σ\n rw [mem_roots, IsRoot.def, eval_map, ← aeval_def, aeval_algHom_apply]\n repeat' simp [minpoly.ne_zero (Algebra.IsSeparable.isIntegral K pb.gen)]\n apply (algebraMap K E).injective\n rw [RingHom.map_mul, RingHom.map_pow, RingHom.map_neg, RingHom.map_one,\n discr_powerBasis_eq_prod'' _ _ _ e]\n congr\n rw [norm_eq_prod_embeddings, prod_prod_Ioi_mul_eq_prod_prod_off_diag]\n conv_rhs =>\n congr\n rfl\n ext σ\n rw [← aeval_algHom_apply,\n aeval_root_derivative_of_splits (minpoly.monic (Algebra.IsSeparable.isIntegral K pb.gen))\n (IsAlgClosed.splits_codomain _) (hroots σ),\n ← Finset.prod_mk _ (hnodup.erase _)]\n rw [Finset.prod_sigma', Finset.prod_sigma']\n refine prod_bij' (fun i _ ↦ ⟨e i.2, e i.1 pb.gen⟩)\n (fun σ hσ ↦ ⟨e.symm (PowerBasis.lift pb σ.2 ?_), e.symm σ.1⟩) ?_ ?_ ?_ ?_ (fun i _ ↦ by simp)\n -- Porting note: `@mem_compl` was not necessary.\n <;> simp only [mem_sigma, mem_univ, Finset.mem_mk, hnodup.mem_erase_iff, IsRoot.def,\n mem_roots',\n mem_singleton, true_and, @mem_compl _ _ _ (_), Sigma.forall, Equiv.apply_symm_apply,\n PowerBasis.lift_gen, implies_true, Equiv.symm_apply_apply,\n Sigma.ext_iff, Equiv.symm_apply_eq, heq_eq_eq, and_true] at *\n · simpa only [aeval_def, eval₂_eq_eval_map] using hσ.2.2\n · exact fun a b hba ↦ ⟨fun h ↦ hba <| e.injective <| pb.algHom_ext h.symm, hroots _⟩\n · rintro a b hba ha\n rw [ha, PowerBasis.lift_gen] at hba\n exact hba.1 rfl\n · exact fun a b _ ↦ pb.algHom_ext <| pb.lift_gen _ _\n\nsection Integral\n\nvariable {R : Type z} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L]\n\n","line":201,"end_line":249,"proof_status":"formalized"} |
| {"id":"lemma:alt_definition_of_trace","label":"alt_definition_of_trace","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(\\alpha \\in K\\) and let \\(\\sigma _i\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Then \\[ \\operatorname {Tr}_{K/\\mathbb {Q}}(\\alpha ) =\\sum _i \\sigma _i(\\alpha ). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/trace_eq_sum_embeddings","latex_link":"sect0002.html#lemma:alt_definition_of_trace","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/trace_eq_sum_embeddings","lean_decl_name":"trace_eq_sum_embeddings","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Trace/Basic.lean","lean4_code":"theorem trace_eq_sum_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L} :\n algebraMap K E (Algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x := by\n have hx := Algebra.IsSeparable.isIntegral K x\n let pb := adjoin.powerBasis hx\n rw [trace_eq_trace_adjoin K x, Algebra.smul_def, RingHom.map_mul, ← adjoin.powerBasis_gen hx,\n trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits_codomain _), ← Algebra.smul_def,\n algebraMap_smul]\n · exact (sum_embeddings_eq_finrank_mul L E pb).symm\n · haveI := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L\n exact Algebra.IsSeparable.isSeparable K _\n\n","line":254,"end_line":264,"proof_status":"formalized"} |
| {"id":"lemma:disc_via_embs","label":"disc_via_embs","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alt_definition_of_trace","defn_of_disc"],"nlp_text":"Let \\(K\\) be a number field with basis \\(B=\\{ b_1,\\dots ,b_n\\} \\) and let \\(\\sigma _1,\\dots ,\\sigma _n\\) be the embeddings of \\(K\\) into \\(\\mathbb {C}\\). Now let \\(M\\) be the matrix \\[ \\left(\\begin{matrix} \\sigma _1(b_1) & \\cdots & \\sigma _1(b_n) \\\\ \\vdots & & \\vdots \\\\ \\sigma _n(b_1) & \\cdots & \\sigma _n(b_n) \\end{matrix} \\right). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","latex_link":"sect0002.html#lemma:disc_via_embs","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","lean_decl_name":"Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_eq_det_embeddingsMatrixReindex_pow_two\n [Algebra.IsSeparable K L] (e : ι ≃ (L →ₐ[K] E)) :\n algebraMap K E (discr K b) = (embeddingsMatrixReindex K E b e).det ^ 2 := by\n rw [discr_def, RingHom.map_det, RingHom.mapMatrix_apply,\n traceMatrix_eq_embeddingsMatrixReindex_mul_trans, det_mul, det_transpose, pow_two]\n\n/-- The discriminant of a power basis. -/\n","line":143,"end_line":149,"proof_status":"formalized"} |
| {"id":"lemma:disc_int_basis","label":"disc_int_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:int_basis_int_disc","lem:disc_change_of_basis"],"nlp_text":"Let \\(K = \\mathbb {Q}(\\alpha )\\) be a number field, where \\(\\alpha \\) is an algebraic integer. Let \\(B = \\{ 1, \\alpha , \\ldots , \\alpha ^{[K : \\mathbb {Q}] - 1} \\} \\) be the basis given by \\(\\alpha \\) and let \\(x \\in \\mathcal{O}_K\\). Then \\(\\Delta (B)x \\in \\mathbb {Z}[\\alpha ]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_mul_isIntegral_mem_adjoin","latex_link":"sect0002.html#lemma:disc_int_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_mul_isIntegral_mem_adjoin","lean_decl_name":"Algebra.discr_mul_isIntegral_mem_adjoin","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_mul_isIntegral_mem_adjoin [Algebra.IsSeparable K L] [IsIntegrallyClosed R]\n [IsFractionRing R K] {B : PowerBasis K L} (hint : IsIntegral R B.gen) {z : L}\n (hz : IsIntegral R z) : discr K B.basis • z ∈ adjoin R ({B.gen} : Set L) := by\n have hinv : IsUnit (traceMatrix K B.basis).det := by\n simpa [← discr_def] using discr_isUnit_of_basis _ B.basis\n have H :\n (traceMatrix K B.basis).det • (traceMatrix K B.basis) *ᵥ (B.basis.equivFun z) =\n (traceMatrix K B.basis).det • fun i => trace K L (z * B.basis i) := by\n congr; exact traceMatrix_of_basis_mulVec _ _\n have cramer := mulVec_cramer (traceMatrix K B.basis) fun i => trace K L (z * B.basis i)\n suffices ∀ i, ((traceMatrix K B.basis).det • B.basis.equivFun z) i ∈ (⊥ : Subalgebra R K) by\n rw [← B.basis.sum_repr z, Finset.smul_sum]\n refine Subalgebra.sum_mem _ fun i _ => ?_\n replace this := this i\n rw [← discr_def, Pi.smul_apply, mem_bot] at this\n obtain ⟨r, hr⟩ := this\n rw [Basis.equivFun_apply] at hr\n rw [← smul_assoc, ← hr, algebraMap_smul]\n refine Subalgebra.smul_mem _ ?_ _\n rw [B.basis_eq_pow i]\n exact Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton _)) _\n intro i\n rw [← H, ← mulVec_smul] at cramer\n replace cramer := congr_arg (mulVec (traceMatrix K B.basis)⁻¹) cramer\n rw [mulVec_mulVec, nonsing_inv_mul _ hinv, mulVec_mulVec, nonsing_inv_mul _ hinv, one_mulVec,\n one_mulVec] at cramer\n rw [← congr_fun cramer i, cramer_apply, det_apply]\n refine\n Subalgebra.sum_mem _ fun σ _ => Subalgebra.zsmul_mem _ (Subalgebra.prod_mem _ fun j _ => ?_) _\n by_cases hji : j = i\n · simp only [updateCol_apply, hji, PowerBasis.coe_basis]\n exact mem_bot.2 (IsIntegrallyClosed.isIntegral_iff.1 <| isIntegral_trace (hz.mul <| hint.pow _))\n · simp only [updateCol_apply, hji, PowerBasis.coe_basis]\n exact mem_bot.2\n (IsIntegrallyClosed.isIntegral_iff.1 <| isIntegral_trace <| (hint.pow _).mul (hint.pow _))\n\nend Integral\n\nend Field\n\nsection Int\n\n/-- Two (finite) ℤ-bases have the same discriminant. -/\n","line":261,"end_line":303,"proof_status":"formalized"} |
| {"id":"theorem:ring_of_ints_of_cyclo","label":"ring_of_ints_of_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:disc_int_basis","lem:discr_of_cyclo","lemma:eis_crit_and_alg_ints"],"nlp_text":"Let \\(\\zeta _p\\) be a \\(p\\)-th root of unity for \\(p\\) an odd prime, let \\(\\lambda _p=1-\\zeta _p\\) and \\(K=\\mathbb {Q}(\\zeta _p)\\). Then \\(\\mathcal{O}_K=\\mathbb {Z}[\\zeta _p]=\\mathbb {Z}[\\lambda _p]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","latex_link":"sect0003.html#theorem:ring_of_ints_of_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","lean_decl_name":"IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/Cyclotomic/Rat.lean","lean4_code":"theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]\n (hζ : IsPrimitiveRoot ζ (p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K := by\n refine ⟨Subtype.val_injective, @fun x => ⟨fun h => ⟨⟨x, ?_⟩, rfl⟩, ?_⟩⟩\n swap\n · rintro ⟨y, rfl⟩\n exact\n IsIntegral.algebraMap\n ((le_integralClosure_iff_isIntegral.1\n (adjoin_le_integralClosure (hζ.isIntegral (NeZero.pos _)))).isIntegral _)\n let B := hζ.subOnePowerBasis ℚ\n have hint : IsIntegral ℤ B.gen := (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one\n -- Porting note: the following `letI` was not needed because the locale `cyclotomic` set it\n -- as instances.\n letI := IsCyclotomicExtension.finiteDimensional {p ^ k} ℚ K\n have H := discr_mul_isIntegral_mem_adjoin ℚ hint h\n obtain ⟨u, n, hun⟩ := discr_prime_pow_eq_unit_mul_pow' hζ\n rw [hun] at H\n replace H := Subalgebra.smul_mem _ H u.inv\n rw [← smul_assoc, ← smul_mul_assoc, Units.inv_eq_val_inv, zsmul_eq_mul, ← Int.cast_mul,\n Units.inv_mul, Int.cast_one, one_mul, smul_def, map_pow] at H\n cases k\n · haveI : IsCyclotomicExtension {1} ℚ K := by simpa using hcycl\n have : x ∈ (⊥ : Subalgebra ℚ K) := by\n rw [singleton_one ℚ K]\n exact mem_top\n obtain ⟨y, rfl⟩ := mem_bot.1 this\n replace h := (isIntegral_algebraMap_iff (algebraMap ℚ K).injective).1 h\n obtain ⟨z, hz⟩ := IsIntegrallyClosed.isIntegral_iff.1 h\n rw [← hz, ← IsScalarTower.algebraMap_apply]\n exact Subalgebra.algebraMap_mem _ _\n · have hmin : (minpoly ℤ B.gen).IsEisensteinAt (Submodule.span ℤ {(p : ℤ)}) := by\n have h₁ := minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hint\n have h₂ := hζ.minpoly_sub_one_eq_cyclotomic_comp (cyclotomic.irreducible_rat (NeZero.pos _))\n rw [IsPrimitiveRoot.subOnePowerBasis_gen] at h₁\n rw [h₁, ← map_cyclotomic_int, show Int.castRingHom ℚ = algebraMap ℤ ℚ by rfl,\n show X + 1 = map (algebraMap ℤ ℚ) (X + 1) by simp, ← map_comp] at h₂\n rw [IsPrimitiveRoot.subOnePowerBasis_gen,\n map_injective (algebraMap ℤ ℚ) (algebraMap ℤ ℚ).injective_int h₂]\n exact cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt p _\n refine\n adjoin_le ?_\n (mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt (n := n)\n (Nat.prime_iff_prime_int.1 hp.out) hint h (by simpa using H) hmin)\n simp only [Set.singleton_subset_iff, SetLike.mem_coe]\n exact Subalgebra.sub_mem _ (self_mem_adjoin_singleton ℤ _) (Subalgebra.one_mem _)\n\n","line":72,"end_line":117,"proof_status":"formalized"} |
| {"id":"lemma:ideals_mult_to_power","label":"ideals_mult_to_power","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(R\\) be a Dedekind domain, \\(p\\) a prime and \\(\\mathfrak {a},\\mathfrak {b},\\mathfrak {c}\\) ideals such that \\[ \\mathfrak {a}\\mathfrak {b}=\\mathfrak {c}^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/ideal.exists_eq_pow_of_mul_eq_pow","latex_link":"sect0003.html#lemma:ideals_mult_to_power","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/ideal.exists_eq_pow_of_mul_eq_pow","lean_decl_name":"ideal.exists_eq_pow_of_mul_eq_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/GCDMonoid/Basic.lean","lean4_code":"theorem exists_eq_pow_of_mul_eq_pow [GCDMonoid α] [Subsingleton αˣ]\n {a b c : α} (hab : IsUnit (gcd a b)) {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, a = d ^ k :=\n let ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow hab h\n ⟨d, (associated_iff_eq.mp hd).symm⟩\n\n","line":590,"end_line":594,"proof_status":"formalized"} |
| {"id":"lemma:may_assume_coprime","label":"may_assume_coprime","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:ideals_mult_to_power","theorem:ring_of_ints_of_cyclo","lemma:unit_lemma","lemma:zeta_pow_sub_eq_unit_zeta_sub_one"],"nlp_text":"Let \\(p \\geq 3\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(K=\\mathbb {Q}(\\zeta _p)\\). Assume that we have \\(x,y,z \\in \\mathbb {Z}\\) with \\(\\gcd (xyz,p)=1\\) and such that \\[ x^p+y^p=z^p. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.may_assume","latex_link":"sect0004.html#lemma:may_assume_coprime","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.may_assume","lean_decl_name":"FltRegular.CaseI.may_assume","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/CaseI/Statement.lean","lean4_code":" rw [← Int.even_pow' two_ne_zero, ← H]\n exact (Odd.of_mul_left (Odd.of_mul_left hI)).pow.add_odd\n (Odd.of_mul_right (Odd.of_mul_left hI)).pow\n have hprod : a * b * c ≠ 0 := by\n intro h\n simp [h] at hI\n have hp5 : 5 ≤ p := by\n by_contra! habs\n have : 2 < p := Nat.lt_of_le_of_ne hpri.out.two_le hodd.symm\n interval_cases p\n · exact fermatLastTheoremFor_iff_int.1 fermatLastTheoremThree a b c\n (fun ha ↦ hprod <| by simp [ha]) (fun hb ↦ hprod <| by simp [hb])\n (fun hc ↦ hprod <| by simp [hc]) H\n · exact Nat.not_prime_mul one_lt_two.ne' one_lt_two.ne' hpri.out\n let d := ({a, b, c} : Finset ℤ).gcd id\n have hd : d ≠ 0 := Finset.gcd_ne_zero_iff.mpr ⟨c, by simp, fun hc ↦ hprod <| by simp_all⟩\n have hdiv : ¬↑p ∣ a / d * (b / d) * (c / d) := by\n contrapose! hI with hdiv\n have hadiv : d ∣ a := gcd_dvd (by simp)\n have hbdiv : d ∣ b := gcd_dvd (by simp)\n have hcdiv : d ∣ c := gcd_dvd (by simp)\n rw [← Int.ediv_mul_cancel hadiv, ← Int.ediv_mul_cancel hbdiv, ← Int.ediv_mul_cancel hcdiv]\n convert dvd_mul_of_dvd_right hdiv (d * d * d) using 1\n grind\n rcases MayAssume.coprime H hprod with ⟨Hxyz, hunit, hprodxyx⟩\n obtain ⟨X, Y, Z, H1, H2, H3, _, H5⟩ := a_not_cong_b hpri.out hp5 hprodxyx Hxyz hunit hdiv\n exact Heasy hreg hp5 H2 H3 H5 H1\n\nend CaseI\n\ntheorem ab_coprime {a b c : ℤ} (H : a ^ p + b ^ p = c ^ p) (hpzero : p ≠ 0)\n (hgcd : ({a, b, c} : Finset ℤ).gcd id = 1) : IsCoprime a b := by\n rw [isCoprime_iff_gcd_eq_one]\n by_contra! h\n obtain ⟨q, hqpri, hq⟩ := exists_prime_and_dvd h\n replace hqpri : Prime (q : ℤ) := prime_iff_natAbs_prime.2 (by simp [hqpri])\n obtain ⟨n, hn⟩ := hq\n have haq : ↑q ∣ a := by\n","line":33,"end_line":70,"proof_status":"formalized"} |
| {"id":"lem:dvd_coeff_cycl_integer","label":"dvd_coeff_cycl_integer","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\( \\alpha \\in \\mathbb {Z}[\\zeta _p]\\) with \\(\\alpha =\\sum _i a_i \\zeta _p^i\\). Let us suppose that there is \\(i\\) such that \\(a_i = 0\\). If \\(n\\) is an integer that divides \\(\\alpha \\) in \\(\\mathbb {Z}[\\zeta _p]\\), then \\(n\\) divides each \\(a_i\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/dvd_coeff_cycl_integer","latex_link":"sect0004.html#lem:dvd_coeff_cycl_integer","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/dvd_coeff_cycl_integer","lean_decl_name":"dvd_coeff_cycl_integer","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":"\nend IntFacts\n","line":377,"end_line":423,"proof_status":"formalized"} |
| {"id":"theorem:FLT_case_one","label":"FLT_case_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:dvd_coeff_cycl_integer","lem:exists_int_sum_eq_zero","lem:flt_ideals_coprime","defn:is_regular_number","lemma:may_assume_coprime"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseI","latex_link":"sect0004.html#theorem:FLT_case_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseI","lean_decl_name":"FltRegular.caseI","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:gen_ideal_coprimality","label":"gen_ideal_coprimality","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(x,y,z,\\epsilon \\in \\mathbb {Z}[\\zeta _p]\\), \\(\\epsilon \\) a unit, and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Assume \\(x,y,z\\) are coprime to \\((1-\\zeta _p)\\), \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\) and \\(x+y\\) is divisible by \\(\\mathfrak {p}^2\\) and \\(x+ \\zeta _p^ky\\) is only divisible by \\(\\mathfrak {p}=(1-\\zeta _p)\\) (for \\(0 \\lt k \\leq p-1\\)). Let \\(\\mathfrak {m}= \\mathrm{gcd}((x),(y))\\). Then: We can write \\[ (x+y)=\\mathfrak {p}^{p(m-1)+1}\\mathfrak {m}\\mathfrak {c}_0 \\]","doc_url":null,"latex_link":"sect0004.html#lem:gen_ideal_coprimality","lean_link":null,"lean_decl_name":"gen_ideal_coprimality","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"thm:gen_flt_eqn","label":"gen_flt_eqn","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:gen_ideal_coprimality","lem:gen_dvd_by_frak_p"],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(\\epsilon \\in \\mathbb {Z}[\\zeta _p]^\\times \\) and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Then the equation \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\) has no solutions with \\(x,y,z \\in \\mathbb {Z}[\\zeta _p]\\), all non-zero and \\(xyz\\) coprime to \\((1-\\zeta _p)\\).","doc_url":null,"latex_link":"sect0004.html#thm:gen_flt_eqn","lean_link":null,"lean_decl_name":"gen_flt_eqn","kind_description":"theorems and lemmas","status_from_color":"statement_formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"defn_of_disc","label":"defn_of_disc","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(A,K\\) be commutative rings with \\(K\\) and \\(A\\)-algebra. let \\(B=\\{ b_1,\\dots ,b_n\\} \\) be a set of elements in \\(K\\). The discriminant of \\(B\\) is defined as \\[ \\Delta (B)= \\det \\left(\\begin{matrix} \\operatorname {Tr}_{K/A}(b_1b_1) & \\cdots & \\operatorname {Tr}_{K/A}(b_1b_n) \\\\ \\vdots & & \\vdots \\\\ \\operatorname {Tr}_{K/A}(b_nb_1) & \\cdots & \\operatorname {Tr}_{K/A}(b_nb_n) \\end{matrix} \\right). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr","latex_link":"sect0002.html#defn_of_disc","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr","lean_decl_name":"Algebra.discr","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"noncomputable def discr (A : Type u) {B : Type v} [CommRing A] [CommRing B] [Algebra A B]\n [Fintype ι] (b : ι → B) := (traceMatrix A b).det\n\n","line":66,"end_line":68,"proof_status":"formalized"} |
| {"id":"lemma:disc_of_prim_elt_basis","label":"disc_of_prim_elt_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc","lemma:disc_via_embs"],"nlp_text":"Let \\(K\\) be a number field and \\(B=\\{ 1,\\alpha ,\\alpha ^2,\\dots ,\\alpha ^{n-1}\\} \\) for some \\(\\alpha \\in K\\). Then \\[ \\Delta (B)=\\prod _{i \\lt j} (\\sigma _i(\\alpha )-\\sigma _j(\\alpha ))^2 \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_prod","latex_link":"sect0002.html#lemma:disc_of_prim_elt_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_powerBasis_eq_prod","lean_decl_name":"Algebra.discr_powerBasis_eq_prod","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_powerBasis_eq_prod (e : Fin pb.dim ≃ (L →ₐ[K] E)) [Algebra.IsSeparable K L] :\n algebraMap K E (discr K pb.basis) =\n ∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) ^ 2 := by\n rw [discr_eq_det_embeddingsMatrixReindex_pow_two K E pb.basis e,\n embeddingsMatrixReindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow]\n congr; ext i\n rw [← prod_pow]\n\n/-- A variation of `Algebra.discr_powerBasis_eq_prod`. -/\n","line":150,"end_line":158,"proof_status":"formalized"} |
| {"id":"lemma:int_basis_int_disc","label":"int_basis_int_disc","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc","lemma:trace_of_alg_int_is_int"],"nlp_text":"Let \\(K\\) be a number field and \\(B=\\{ b_1,\\dots ,b_n\\} \\) be elements in \\(\\mathcal{O}_K\\), then \\(\\Delta (B) \\in \\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_isIntegral","latex_link":"sect0002.html#lemma:int_basis_int_disc","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_isIntegral","lean_decl_name":"Algebra.discr_isIntegral","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_isIntegral {b : ι → L} (h : ∀ i, IsIntegral R (b i)) : IsIntegral R (discr K b) := by\n classical\n rw [discr_def]\n exact IsIntegral.det fun i j ↦ isIntegral_trace ((h i).mul (h j))\n\n/-- Let `K` be the fraction field of an integrally closed domain `R` and let `L` be a finite\nseparable extension of `K`. Let `B : PowerBasis K L` be such that `IsIntegral R B.gen`.\nThen for all, `z : L` that are integral over `R`, we have\n`(discr K B.basis) • z ∈ adjoin R ({B.gen} : Set L)`. -/\n","line":252,"end_line":260,"proof_status":"formalized"} |
| {"id":"lem:disc_change_of_basis","label":"disc_change_of_basis","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["defn_of_disc"],"nlp_text":"Let \\(K\\) be a number field and \\(B,B'\\) bases for \\(K/\\mathbb {Q}\\). If \\(P\\) denotes the change of basis matrix, then \\[ \\Delta (B)=\\det (P)^2 \\Delta (B'). \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_of_matrix_mulVec","latex_link":"sect0002.html#lem:disc_change_of_basis","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.discr_of_matrix_mulVec","lean_decl_name":"Algebra.discr_of_matrix_mulVec","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Discriminant.lean","lean4_code":"theorem discr_of_matrix_mulVec (b : ι → B) (P : Matrix ι ι A) :\n discr A (P.map (algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b := by\n rw [discr_def, traceMatrix_of_matrix_mulVec, det_mul, det_mul, det_transpose, mul_comm, ←\n mul_assoc, discr_def, pow_two]\n\nend Basic\n\nsection Field\n\nvariable (K : Type u) {L : Type v} (E : Type z) [Field K] [Field L] [Field E]\nvariable [Algebra K L] [Algebra K E]\nvariable [Module.Finite K L] [IsAlgClosed E]\n\n/-- If `b` is a basis of a finite separable field extension `L/K`, then `Algebra.discr K b ≠ 0`. -/\n","line":113,"end_line":126,"proof_status":"formalized"} |
| {"id":"lem:exists_int_sum_eq_zero","label":"exists_int_sum_eq_zero","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:exists_int_sub_pow_prime_dvd","lemma:unit_lemma"],"nlp_text":"Let \\(p \\geq 3\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\( \\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Let \\(x\\) and \\(y\\) be integers such that \\(x+y\\zeta _p^i=u \\alpha ^p\\) with \\(u \\in \\mathbb {Z}[\\zeta _p]^\\times \\) and \\(\\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Then there is an integer \\(k\\) such that \\[ x+y\\zeta _p^i-\\zeta _p^{2k}x-\\zeta _p^{2k-i}y \\equiv 0 \\pmod p. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.exists_int_sum_eq_zero","latex_link":"sect0004.html#lem:exists_int_sum_eq_zero","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.CaseI.exists_int_sum_eq_zero","lean_decl_name":"FltRegular.CaseI.exists_int_sum_eq_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CaseI.lean","lean4_code":" ringOfIntegersComplexConj K (x + y * ↑(hζ.unit' ^ i) : 𝓞 K) =\n x + y * (hζ.unit' ^ (-i) : (𝓞 K)ˣ) := by\n haveI := IsCyclotomicExtension.IsCMField K hp\n ext1\n simp only [map_add, map_intCast, map_mul, coe_ringOfIntegersComplexConj, zpow_neg, map_units_inv,\n add_right_inj, mul_eq_mul_left_iff, Int.cast_eq_zero]\n simp_rw [NumberField.Units.coe_zpow]\n left\n simp only [map_zpow₀]\n rw [← inv_zpow]\n congr\n suffices hζ.unit' ∈ Units.torsion K by\n","line":30,"end_line":41,"proof_status":"formalized"} |
| {"id":"lemma:cyclo_poly_deg","label":"cyclo_poly_deg","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(n\\) any integer, \\(\\Phi _n\\) (the \\(n\\)-th cyclotomic polynomial) is a polynomial of degree \\(\\varphi (n)\\) (where \\(\\varphi \\) is Euler’s Totient function).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.degree_cyclotomic","latex_link":"sect0003.html#lemma:cyclo_poly_deg","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.degree_cyclotomic","lean_decl_name":"Polynomial.degree_cyclotomic","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean","lean4_code":"theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :\n (cyclotomic' n R).degree = Nat.totient n := by\n simp only [degree_eq_natDegree (cyclotomic'_ne_zero n R), natDegree_cyclotomic' h]\n\n/-- The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. -/\n","line":105,"end_line":109,"proof_status":"formalized"} |
| {"id":"lem:discr_of_cyclo","label":"discr_of_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:cyclo_poly_deg","lemma:num_field_disc_in_terms_of_norm","lemma:cyclo_poly_irr"],"nlp_text":"Let \\(\\zeta _p\\) be a \\(p\\)-th root of unity for \\(p\\) an odd prime, let \\(\\lambda _p=1-\\zeta _p\\) and \\(K=\\mathbb {Q}(\\zeta _p)\\). Then \\[ \\Delta (\\{ 1,\\zeta _p,\\dots ,\\zeta _p^{p-2}\\} )=\\Delta (\\{ 1,\\lambda _p,\\dots ,\\lambda _p^{p-2}\\} )=(-1)^{\\frac{(p-1)}{2}}p^{p-2}. \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.discr_prime_pow'","latex_link":"sect0003.html#lem:discr_of_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsCyclotomicExtension.Rat.discr_prime_pow'","lean_decl_name":"IsCyclotomicExtension.Rat.discr_prime_pow'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:exists_int_sub_pow_prime_dvd","label":"exists_int_sub_pow_prime_dvd","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(\\alpha \\in \\mathbb {Z}[\\zeta _p]\\). Then \\(\\alpha ^p\\) is congruent to an integer modulo \\(p\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_int_sub_pow_prime_dvd","latex_link":"sect0004.html#lem:exists_int_sub_pow_prime_dvd","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_int_sub_pow_prime_dvd","lean_decl_name":"exists_int_sub_pow_prime_dvd","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":" isPrimitiveRoot_of_mem_primitiveRoots <| mem_primitiveRoots_of_mem_nthRootsFinset hp hη hne1\n\nend Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\n\nopen FiniteDimensional Polynomial Algebra Nat Finset Fintype\n\nvariable (p : ℕ) (L : Type*) [Field L] [CharZero L] [IsCyclotomicExtension {p} ℚ L]\n\ninstance Ring.toSubtractionMonoid {S : Type*} [Ring S] : SubtractionMonoid S := inferInstance\n\nsection IntFacts\n\nopen scoped NumberField\n\ninstance {K : Type*} [Field K] : Module (𝓞 K) (𝓞 K) := Semiring.toModule\n\nopen Ideal IsCyclotomicExtension\n\ntheorem exists_int_sub_pow_prime_dvd {A : Type*} [CommRing A] [IsCyclotomicExtension {p} ℤ A]\n [hp : Fact p.Prime] (a : A) : ∃ m : ℤ, a ^ p - m ∈ span ({(p : A)} : Set A) := by\n refine Algebra.adjoin_induction ?_ ?_ ?_ ?_ (adjoin_roots (S := {p}) (A := ℤ) a)\n · intro x hx\n use 1\n simp_all\n · intro r\n use r ^ p\n simp\n · rintro x y _ _ ⟨b, hb⟩ ⟨c, hc⟩\n obtain ⟨r, hr⟩ := exists_add_pow_prime_eq hp.out x y\n rw [mul_assoc, mul_assoc] at hr\n rw [hr]\n use c + b\n push_cast\n rw [sub_add_eq_sub_sub, sub_eq_add_neg, sub_eq_add_neg, add_comm (x ^ p + _), add_assoc,\n add_assoc]\n apply Ideal.add_mem _ _\n","line":27,"end_line":62,"proof_status":"formalized"} |
| {"id":"lem:loc_ramification","label":"loc_ramification","background_color":"#a3d6ff","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number field, \\(p\\) be a rational prime below \\(\\mathfrak {p}\\) and let \\(\\alpha _0, \\xi \\) be in \\(\\mathcal{O}_\\mathfrak {p}^\\times \\) (the units of the ring of integers of the completion of \\(K\\) at \\(\\mathfrak {p}\\)) be such that \\[ \\alpha _0^p \\equiv \\xi \\mod \\mathfrak {p}^{m+r} \\]","doc_url":null,"latex_link":"sect0006.html#lem:loc_ramification","lean_link":null,"lean_decl_name":"loc_ramification","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:ramification_lem","label":"ramification_lem","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:loc_ramification","lem:diff_ideal_eqn","lem:diff_ram"],"nlp_text":"Let \\(F=\\mathbb {Q}(\\zeta _p)\\) with \\(\\zeta _p\\) a primitive \\(p\\)-th root of unity. If \\(\\xi \\in \\mathcal{O}_F\\) and coprime to \\(\\lambda _p:=1-\\zeta _p\\) and if \\[ \\xi \\equiv \\alpha _0^p \\mod \\lambda _p^p \\]","doc_url":null,"latex_link":"sect0006.html#lem:ramification_lem","lean_link":null,"lean_decl_name":"ramification_lem","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"theorem:FLT_case_two","label":"FLT_case_two","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["thm:gen_flt_eqn","thm:Kummers_lemma"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseII","latex_link":"sect0004.html#theorem:FLT_case_two","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/FltRegular.caseII","lean_decl_name":"FltRegular.caseII","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/CaseII/Statement.lean","lean4_code":"section Mathlib.Data.Int.Init\n\nlemma Int.gcd_left_comm (a b c : ℤ) : Int.gcd a (Int.gcd b c) = Int.gcd b (Int.gcd a c) := by\n rw [← Int.gcd_assoc, ← Int.gcd_assoc, Int.gcd_comm a b]\n\nend Mathlib.Data.Int.Init\n\n/-- CaseII. -/\ntheorem caseII {a b c : ℤ} {p : ℕ} [hpri : Fact p.Prime] (hreg : IsRegularPrime p) (hodd : p ≠ 2)\n (hprod : a * b * c ≠ 0) (hgcd : ({a, b, c} : Finset ℤ).gcd id = 1)\n (caseII : ↑p ∣ a * b * c) : a ^ p + b ^ p ≠ c ^ p := by\n intro e\n simp only [ne_eq, mul_eq_zero, not_or] at hprod\n obtain ⟨⟨a0, b0⟩, c0⟩ := hprod\n have hodd' := Nat.Prime.odd_of_ne_two hpri.out hodd\n obtain hab | hc := (Nat.prime_iff_prime_int.mp hpri.out).dvd_or_dvd caseII\n · obtain ha| hb := (Nat.prime_iff_prime_int.mp hpri.out).dvd_or_dvd hab\n · refine not_exists_Int_solution' hreg hodd ⟨b, -c, -a, ?_, ?_, ?_, ?_⟩\n · simp only [← hgcd, Finset.gcd_insert,id_eq, ← Int.coe_gcd, Int.neg_gcd,\n ← LawfulSingleton.insert_empty_eq, Finset.gcd_empty, Int.gcd_left_comm _ a]\n · rwa [dvd_neg]\n · rwa [ne_eq, neg_eq_zero]\n · simp [hodd'.neg_pow, ← e]\n · refine not_exists_Int_solution' hreg hodd ⟨-c, a, -b, ?_, ?_, ?_, ?_⟩\n","line":80,"end_line":103,"proof_status":"formalized"} |
| {"id":"FLT_regular","label":"FLT_regular","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["theorem:FLT_case_two","theorem:FLT_case_one"],"nlp_text":"Let \\(p\\) be an odd regular prime. Then \\[ x^p+y^p=z^p \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/flt_regular","latex_link":"sect0004.html#FLT_regular","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/flt_regular","lean_decl_name":"flt_regular","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/FltRegular.lean","lean4_code":"theorem flt_regular {p : ℕ} [Fact p.Prime] (hreg : IsRegularPrime p) (hodd : p ≠ 2) :\n FermatLastTheoremFor p := by\n apply fermatLastTheoremFor_iff_int.mpr\n intro a b c ha hb hc e\n have hprod := mul_ne_zero (mul_ne_zero ha hb) hc\n obtain ⟨e', hgcd, hprod'⟩ := MayAssume.coprime e hprod\n let d := ({a, b, c} : Finset ℤ).gcd id\n by_cases case : ↑p ∣ (a / d) * (b / d) * (c / d)\n · exact caseII hreg hodd hprod' hgcd case e'\n · exact caseI hreg case e'\n","line":8,"end_line":17,"proof_status":"formalized"} |
| {"id":"lemma:unit_lemma","label":"unit_lemma","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:alg_int_abs_val_one","lem:roots_of_unity_in_cyclo"],"nlp_text":"Any unit \\(u\\) in \\(\\mathbb {Z}[\\zeta _p]\\) can be written in the form \\(\\beta \\zeta _p^k \\) with \\(k\\) an integer and \\(\\beta \\in \\mathbb {R}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/unit_lemma_gal_conj","latex_link":"sect0003.html#lemma:unit_lemma","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/unit_lemma_gal_conj","lean_decl_name":"unit_lemma_gal_conj","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:flt_ideals_coprime","label":"flt_ideals_coprime","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:zeta_pow_sub_eq_unit_zeta_sub_one"],"nlp_text":"Let \\(p \\geq 5\\) be an prime number, \\(\\zeta _p\\) a \\(p\\)-th root of unity and \\(x, y \\in \\mathbb {Z}\\) coprime. For \\(i \\neq j\\) with \\(i,j \\in {0,\\dots ,p-1}\\) we can write \\[ (\\zeta _p^i-\\zeta _p^j)=u(1-\\zeta _p) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/fltIdeals_coprime","latex_link":"sect0004.html#lem:flt_ideals_coprime","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/fltIdeals_coprime","lean_decl_name":"fltIdeals_coprime","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/CyclRat.lean","lean4_code":" ← (Fin.castOrderIso hdim).toEquiv.sum_comp] at hy\n simp only [RelIso.coe_fn_toEquiv, Fin.val_castSucc, Fin.coe_orderIso_apply] at hy\n conv_lhs at hy =>\n congr; rfl; ext x\n rw [smul_neg]\n congr; congr; rfl; congr\n rw [show ζ = hζ'.toInteger from rfl, ← hζ'.integralPowerBasis_gen, ← hb]\n rfl; rfl; congr; congr; rfl; congr\n rw [show ζ = hζ'.toInteger from rfl, ← hζ'.integralPowerBasis_gen, ← hb]\n conv_lhs at hy =>\n congr; rfl; ext x\n rw [← show ∀ y, _ = _ from fun y => congr_fun b.coe_basis y, ← sub_eq_add_neg]\n rw [sum_sub_distrib] at hy\n replace hy := congr_arg (b.basis.coord ((Fin.castOrderIso hdim.symm) ⟨i, hi⟩)) hy\n rw [← b.basis.equivFun_symm_apply, ← b.basis.equivFun_symm_apply, LinearMap.map_sub,\n b.basis.coord_equivFun_symm, b.basis.coord_equivFun_symm, ← smul_eq_mul,\n","line":308,"end_line":323,"proof_status":"formalized"} |
| {"id":"Kummer_alt","label":"Kummer_alt","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:ramification_lem","Hilbert90","lem:Hilbert92"],"nlp_text":"Let \\(u \\in F\\) with \\(F=\\mathbb {Q}(\\zeta _p)\\) be a unit such that \\[ u \\equiv a^p \\mod \\lambda _p^p \\]","doc_url":null,"latex_link":"sect0007.html#Kummer_alt","lean_link":null,"lean_decl_name":"Kummer_alt","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lemma:diff_of_irr_pol","label":"diff_of_irr_pol","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\) be a monic irreducible polynomial over a number field \\(K\\) and let \\(\\alpha \\) be one of its roots in \\(\\mathbb {C}\\). Then \\[ f'(\\alpha )=\\prod _{\\beta \\neq \\alpha } (\\alpha -\\beta ), \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.aeval_root_derivative_of_splits","latex_link":"sect0002.html#lemma:diff_of_irr_pol","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.aeval_root_derivative_of_splits","lean_decl_name":"Polynomial.aeval_root_derivative_of_splits","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Algebra/Polynomial/Splits.lean","lean4_code":"theorem aeval_root_derivative_of_splits [Algebra K L] [DecidableEq L] {P : K[X]} (hmo : P.Monic)\n (hP : P.Splits (algebraMap K L)) {r : L} (hr : r ∈ P.aroots L) :\n aeval r (Polynomial.derivative P) =\n (((P.aroots L).erase r).map fun a => r - a).prod := by\n replace hmo := hmo.map (algebraMap K L)\n replace hP := (splits_id_iff_splits (algebraMap K L)).2 hP\n rw [aeval_def, ← eval_map, ← derivative_map]\n nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]\n rw [eval_multiset_prod_X_sub_C_derivative hr]\n\n/-- If `P` is a monic polynomial that splits, then `coeff P 0` equals the product of the roots. -/\n","line":537,"end_line":547,"proof_status":"formalized"} |
| {"id":"lemma:alg_int_abs_val_one","label":"alg_int_abs_val_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\alpha \\) be an algebraic integer all of whose conjugates have absolute value one. Then \\(\\alpha \\) is a root of unity.","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_roots_of_unity_of_abs_eq_one","latex_link":"sect0003.html#lemma:alg_int_abs_val_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_roots_of_unity_of_abs_eq_one","lean_decl_name":"mem_roots_of_unity_of_abs_eq_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"thm:Kummers_lemma","label":"Kummers_lemma","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["Kummer_alt"],"nlp_text":"Let \\(p\\) be a regular prime and let \\(u \\in \\mathbb {Z}[\\zeta _p]^\\times \\). If \\(u \\equiv a \\mod p\\) for some \\(a \\in \\mathbb {Z}\\), then there exists \\(v \\in \\mathbb {Z}[\\zeta _p]^\\times \\) such that \\(u=v^p\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/eq_pow_prime_of_unit_of_congruent","latex_link":"sect0004.html#thm:Kummers_lemma","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/eq_pow_prime_of_unit_of_congruent","lean_decl_name":"eq_pow_prime_of_unit_of_congruent","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/KummersLemma/KummersLemma.lean","lean4_code":" refine Int.modEq_iff_dvd.mp (Int.ModEq.pow_card_sub_one_eq_one hpri.out ?_).symm\n rw [isCoprime_comm, (Nat.prime_iff_prime_int.mp hpri.out).coprime_iff_not_dvd]\n intro h\n replace h := Int.cast_dvd_cast (α := 𝓞 K) _ _ h\n simp only [Int.cast_natCast, ← dvd_iff_dvd_of_dvd_sub hn] at h\n refine hζ.zeta_sub_one_prime'.not_unit ((isUnit_pow_iff ?_).mp\n (isUnit_of_dvd_unit ((associated_zeta_sub_one_pow_prime hζ).dvd.trans h) u.isUnit))\n simpa only [ne_eq, tsub_eq_zero_iff_le, not_le] using hpri.out.one_lt\n replace hn' := Int.cast_dvd_cast (α := 𝓞 K) _ _ hn'\n simp only [Int.cast_natCast, Int.cast_sub, Int.cast_pow, Int.cast_one] at hn'\n rw [← Ideal.mem_span_singleton, ← Ideal.Quotient.eq_zero_iff_mem,\n RingHom.map_sub, sub_eq_zero] at hn hn' ⊢\n rw [Units.val_pow_eq_pow_val, RingHom.map_pow, hn, ← RingHom.map_pow, hn']\n have : (hζ.unit' - 1 : 𝓞 K) ^ p ∣ (↑(u ^ (p - 1)) : 𝓞 K) - 1 := by\n rw [hx]\n rw [sub_eq_iff_eq_add, add_comm] at hx\n have H : Algebra.norm ℤ (1 + p • x) = 1 := norm_add_one_smul_of_isUnit hpri.out\n hp x (by rw [nsmul_eq_mul, ← hx]; exact Units.isUnit _)\n have := H ▸ zeta_sub_one_pow_dvd_norm_sub_pow hζ x\n simpa [ge_iff_le, Int.cast_one, sub_self, nsmul_eq_mul, Nat.cast_mul, PNat.pos,\n Nat.cast_pred, zero_sub, IsUnit.mul_iff, ne_eq, tsub_eq_zero_iff_le, not_le, dvd_neg,\n Units.isUnit, and_true, zero_add] using this\n obtain ⟨v, hv⟩ := exists_pow_eq_of_zeta_sub_one_pow_dvd_sub_one hp hreg hζ this\n have hv' : IsIntegral ℤ v := by\n apply IsIntegral.of_pow (NeZero.pos p); rw [hv]\n exact NumberField.RingOfIntegers.isIntegral_coe _\n set w : 𝓞 K := ⟨v, hv'⟩\n have : IsUnit w := by\n rw [← isUnit_pow_iff (NeZero.pos p).ne.symm]; convert (u ^ (p - 1) : (𝓞 K)ˣ).isUnit; ext\n exact hv\n have hv'' : this.unit ^ p = u ^ (p - 1) := by\n ext; simpa only [Units.val_pow_eq_pow_val, IsUnit.unit_spec, SubmonoidClass.coe_pow] using hv\n use u / this.unit\n rw [div_pow, hv'', div_eq_mul_inv, ← pow_sub _ tsub_le_self,\n tsub_tsub_cancel_of_le (Nat.Prime.one_lt hpri.out).le, pow_one]\n","line":48,"end_line":90,"proof_status":"formalized"} |
| {"id":"Hilbert90","label":"Hilbert90","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lem:exists_alg_int"],"nlp_text":"Let \\(K/F\\) be a Galois extension of number fields whose Galois group \\(\\operatorname{Gal}(K/F)\\) is cyclic with generator \\(\\sigma \\). If \\(\\alpha \\in K\\) is such that \\(N_{K/F}(\\alpha )=1\\), then \\[ \\alpha =\\beta / \\sigma (\\beta ) \\]","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert90","latex_link":"sect0005.html#Hilbert90","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert90","lean_decl_name":"Hilbert90","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:Hilbert92","label":"Hilbert92","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K/F\\) be a Galois extension of \\(F=\\mathbb {Q}(\\zeta _p)\\) with Galois group \\(\\operatorname{Gal}(K/F)\\) cyclic with generator \\(\\sigma \\). Then there exists a unit \\(\\eta \\in \\mathcal{O}_K\\) such that \\(N_{K/F}(\\eta )=1\\) but does not have the form \\(\\epsilon /\\sigma (\\epsilon )\\) for any unit \\(\\epsilon \\in \\mathcal{O}_K\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert92","latex_link":"sect0005.html#lem:Hilbert92","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Hilbert92","lean_decl_name":"Hilbert92","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"./FltRegular/NumberTheory/Hilbert92.lean","lean4_code":"lemma Hilbert92 [Algebra k K] [IsGalois k K] [NumberField k] [NumberField K]\n (hKL : Nat.Prime (finrank k K)) (hpodd : finrank k K ≠ 2)\n (σ : K ≃ₐ[k] K) (hσ : ∀ x, x ∈ Subgroup.zpowers σ) :\n ∃ η : (𝓞 K)ˣ, Algebra.norm k (η : K) = 1 ∧ ∀ ε : (𝓞 K)ˣ, (η : K) ≠ ε / (σ ε : K) :=\n have := IsUnramifiedAtInfinitePlaces_of_odd_finrank (hKL.odd_of_ne_two hpodd)\n have : IsCyclic (K ≃ₐ[k] K) := ⟨σ, hσ⟩\n almostHilbert92 (finrank k K) hKL rfl σ hσ hpodd\n\n","line":867,"end_line":874,"proof_status":"formalized"} |
| {"id":"defn:is_regular_number","label":"is_regular_number","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A prime number \\(p\\) is called regular if it does not divide the class number of \\(\\mathbb {Q}(\\zeta _p)\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsRegularNumber","latex_link":"sect0004.html#defn:is_regular_number","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/IsRegularNumber","lean_decl_name":"IsRegularNumber","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/RegularPrimes.lean","lean4_code":" (L : Type*) [Field L] [CharZero L] [IsCyclotomicExtension {2} ℚ L] :\n IsPrincipalIdealRing (𝓞 L) := by\n","line":51,"end_line":52,"proof_status":"formalized"} |
| {"id":"lem:roots_of_unity_in_cyclo","label":"roots_of_unity_in_cyclo","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a prime, \\(K=\\mathbb {Q}(\\zeta _p)\\) \\(\\alpha \\in K\\) such that there exists \\(n \\in \\mathbb {N}\\) such that \\(\\alpha ^n=1\\), then \\(\\alpha =\\pm \\zeta _p^k\\) for some \\(k\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/roots_of_unity_in_cyclo","latex_link":"sect0003.html#lem:roots_of_unity_in_cyclo","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/roots_of_unity_in_cyclo","lean_decl_name":"roots_of_unity_in_cyclo","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"flt-regular","source_path":"FltRegular/NumberTheory/Cyclotomic/UnitLemmas.lean","lean4_code":" apply_fun (fun n : ℕ => (n : 𝓞 K)) at hk\n rw [Nat.cast_add, Nat.cast_mul, Nat.cast_two, Nat.cast_one, add_comm] at hk\n intro h2m\n have := Ideal.sub_mem I hpm (Ideal.mul_mem_right (↑k) I h2m)\n rw [sub_eq_of_eq_add hk] at this\n exact hζ.isPrime_one_sub_zeta.ne_top (Ideal.eq_top_of_isUnit_mem I this isUnit_one)\n\nlemma neg_one_eq_one_iff_two_eq_zero {M : Type*} [AddGroupWithOne M] :\n (-1 : M) = 1 ↔ (2 : M) = 0 := by rw [neg_eq_iff_add_eq_zero, one_add_one_eq_two]\n\nlemma Units.coe_map_inv' {M N F : Type*} [Monoid M] [Monoid N] [FunLike F M N]\n [MonoidHomClass F M N] (f : F) (m : Mˣ) :\n ↑((Units.map (f : M →* N) m)⁻¹) = f ↑(m⁻¹ : Mˣ) :=\n m.coe_map_inv (f : M →* N)\n\nvariable (K) in\ntheorem IsCyclotomicExtension.IsCMField (hp : 2 < p) :\n IsCMField K :=\n haveI := IsCyclotomicExtension.isAbelianGalois {p} ℚ K\n haveI := nrRealPlaces_eq_zero_iff.1 (Rat.nrRealPlaces_eq_zero K hp)\n ⟨⟩\n\nlemma unit_inv_conj_not_neg_zeta_runity_aux (u : (𝓞 K)ˣ) [Fact (p.Prime)] (hp : 2 < p) :\n haveI := IsCyclotomicExtension.IsCMField K hp\n algebraMap (𝓞 K) (𝓞 K ⧸ I) (unitsMulComplexConjInv K u).1 = 1 := by\n haveI := IsCyclotomicExtension.IsCMField K hp\n have := Units.coe_map_inv' (N := 𝓞 K ⧸ I) (algebraMap (𝓞 K) (𝓞 K ⧸ I)) (unitsComplexConj K u)\n rw [unitsMulComplexConjInv_apply, Units.val_mul, map_mul, ← this, Units.mul_inv_eq_one,\n Units.coe_map , MonoidHom.coe_coe]\n haveI := Fact.mk hp\n have hu := hζ.integralPowerBasis.basis.sum_repr u\n let a := hζ.integralPowerBasis.basis.repr\n let φn := hζ.integralPowerBasis.dim\n simp_rw [PowerBasis.basis_eq_pow, IsPrimitiveRoot.integralPowerBasis_gen] at hu\n have hu' := congr_arg (ringOfIntegersComplexConj K) hu\n replace hu' : ∑ x : Fin φn, (a u) x • (ringOfIntegersComplexConj K)\n (⟨ζ, hζ.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ)) = unitsComplexConj K u := by\n refine Eq.trans ?_ hu'\n rw [map_sum]\n congr 1\n ext x\n congr 1\n rw [map_zsmul]\n have : ∀ x : Fin φn, ringOfIntegersComplexConj K (⟨ζ, hζ.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ)) =\n ⟨ζ⁻¹, hζ.inv.isIntegral (NeZero.pos p)⟩ ^ (x : ℕ) := by\n intro x\n ext\n simp only [map_pow, coe_ringOfIntegersComplexConj, RingOfIntegers.map_mk, inv_pow]\n suffices hζ.unit' ∈ Units.torsion K by\n have H := RingOfIntegers.ext_iff.1 <|\n Units.ext_iff.1 <| unitsComplexConj_torsion K ⟨hζ.unit', ‹_›⟩\n have : ↑↑hζ.unit' = ζ := rfl\n simp only [Units.coe_mapEquiv, AlgEquiv.toRingEquiv_eq_coe, RingEquiv.coe_toMulEquiv,\n","line":206,"end_line":258,"proof_status":"formalized"} |
| {"id":"lem:exists_alg_int","label":"exists_alg_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a number fields and \\(\\alpha \\in K\\). Then there exists a \\(n \\in \\mathbb {Z}\\backslash \\{ 0\\} \\) such that \\(n \\alpha \\) is an algebraic integer.","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_alg_int","latex_link":"sect0005.html#lem:exists_alg_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/exists_alg_int","lean_decl_name":"exists_alg_int","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:diff_ideal_eqn","label":"diff_ideal_eqn","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["def:rel_different"],"nlp_text":"Let \\(K/F\\) be an extension of number fields and let \\(S\\) denote the set of \\(\\alpha \\in \\mathcal{O}_K\\) be such that \\(K=F(\\alpha )\\). Then \\[ \\mathfrak {d}_{K/F} = \\left(m_{\\alpha }'(\\alpha ) : \\alpha \\in S \\right) \\]","doc_url":null,"latex_link":"sect0006.html#lem:diff_ideal_eqn","lean_link":null,"lean_decl_name":"diff_ideal_eqn","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lemma:norm_of_alg_int_is_int","label":"norm_of_alg_int_is_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(\\alpha \\in \\mathcal{O}_K\\) then \\(N_{K/\\mathbb {Q}}(\\alpha )\\) is in \\(\\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_norm","latex_link":"sect0002.html#lemma:norm_of_alg_int_is_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_norm","lean_decl_name":"Algebra.isIntegral_norm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Norm/Basic.lean","lean4_code":"theorem isIntegral_norm [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L]\n [FiniteDimensional K L] {x : L} (hx : IsIntegral R x) : IsIntegral R (norm K x) := by\n have hx' : IsIntegral K x := hx.tower_top\n rw [← isIntegral_algebraMap_iff (algebraMap K (AlgebraicClosure L)).injective, norm_eq_prod_roots]\n · refine (IsIntegral.multiset_prod fun y hy => ?_).pow _\n rw [mem_roots_map (minpoly.ne_zero hx')] at hy\n use minpoly R x, minpoly.monic hx\n rw [← aeval_def] at hy ⊢\n exact minpoly.aeval_of_isScalarTower R x y hy\n · apply IsAlgClosed.splits_codomain\n\n","line":238,"end_line":248,"proof_status":"formalized"} |
| {"id":"lemma:eis_crit_and_alg_ints","label":"eis_crit_and_alg_ints","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":["lemma:norm_of_alg_int_is_int","lemma:trace_of_alg_int_is_int"],"nlp_text":"Let \\(K = \\mathbb {Q}(\\alpha )\\) be a number field, where \\(\\alpha \\) is an algebraic integer with minimal polynomial that is Eisenstein at \\(p\\). Let \\(x \\in \\mathcal{O}_K\\) such that \\(p^n x \\in \\mathbb {Z}[\\alpha ]\\) for some \\(n\\). Then \\(x \\in \\mathbb {Z}[\\alpha ]\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","latex_link":"sect0002.html#lemma:eis_crit_and_alg_ints","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","lean_decl_name":"mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean","lean4_code":"theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}\n (hp : Prime p) (hBint : IsIntegral R B.gen) {n : ℕ} {z : L} (hzint : IsIntegral R z)\n (hz : p ^ n • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :\n z ∈ adjoin R ({B.gen} : Set L) := by\n induction' n with n hn\n · simpa using hz\n · rw [_root_.pow_succ', mul_smul] at hz\n exact\n hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)\n\n","line":360,"end_line":369,"proof_status":"formalized"} |
| {"id":"lemma:zeta_pow_sub_eq_unit_zeta_sub_one","label":"zeta_pow_sub_eq_unit_zeta_sub_one","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be an odd prime, \\(\\zeta _p\\) a primitive \\(p\\)-th root of unity and let \\(K=\\mathbb {Q}(\\zeta _p)\\). Then for any \\(i,j \\in {0,\\dots ,p-1}\\) with \\(i \\ne j\\), there exists a unit \\(u \\in \\mathcal{O}_K^\\times \\) such that \\(\\zeta _p^i-\\zeta _p^j = u (\\zeta _p-1)\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","latex_link":"sect0003.html#lemma:zeta_pow_sub_eq_unit_zeta_sub_one","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","lean_decl_name":"is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lem:diff_ram","label":"diff_ram","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K/F\\) be an extension of number fields with \\(\\mathfrak {p}_F, \\mathfrak {p}_K\\) prime ideas in \\(K,F\\) respectively, with \\(\\mathfrak {p}_K^e \\parallel \\mathfrak {p}_F\\) for \\(e \\gt 0\\). Then \\(\\mathfrak {p}_K^{e-1} \\mid \\mathfrak {d}_{K/F}\\) and \\(\\mathfrak {p}_K^{e} \\mid \\mathfrak {d}_{K/F}\\) iff \\(\\mathfrak {p}_F \\mid e\\).","doc_url":null,"latex_link":"sect0006.html#lem:diff_ram","lean_link":null,"lean_decl_name":"diff_ram","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"def:rel_different","label":"rel_different","background_color":"#b0eca3","border_color":"green","kind":"box","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K, F\\) be number fields with \\(F \\subseteq K\\). Let \\(A\\) be an additive subgroup of \\(K\\). Let \\[ A^{-1}=\\{ \\alpha \\in K | \\alpha A \\in \\mathcal{O}_K\\} \\]","doc_url":null,"latex_link":"sect0006.html#def:rel_different","lean_link":null,"lean_decl_name":"rel_different","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lemma:trace_of_alg_int_is_int","label":"trace_of_alg_int_is_int","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K\\) is a number field and \\(\\alpha \\in \\mathcal{O}_K\\) then \\(\\operatorname {Tr}_{K/\\mathbb {Q}}(\\alpha )\\) is in \\(\\mathbb {Z}\\).","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_trace","latex_link":"sect0002.html#lemma:trace_of_alg_int_is_int","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Algebra.isIntegral_trace","lean_decl_name":"Algebra.isIntegral_trace","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"lemma:cyclo_poly_irr","label":"cyclo_poly_irr","background_color":"#9cec8b","border_color":"green","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(n\\) any integer, \\(\\Phi _n\\) (the \\(n\\)-th cyclotomic polynomial) is an irreducible polynomial .","doc_url":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.cyclotomic.irreducible","latex_link":"sect0003.html#lemma:cyclo_poly_irr","lean_link":"https://cbirkbeck.github.io/FltRegulartest/docs//find/#doc/Polynomial.cyclotomic.irreducible","lean_decl_name":"Polynomial.cyclotomic.irreducible","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/FieldTheory/Minpoly/Basic.lean","lean4_code":"theorem irreducible (hx : IsIntegral A x) : Irreducible (minpoly A x) := by\n refine (irreducible_of_monic (monic hx) <| ne_one A x).2 fun f g hf hg he => ?_\n rw [← hf.isUnit_iff, ← hg.isUnit_iff]\n by_contra! h\n have heval := congr_arg (Polynomial.aeval x) he\n rw [aeval A x, aeval_mul, mul_eq_zero] at heval\n rcases heval with heval | heval\n · exact aeval_ne_zero_of_dvdNotUnit_minpoly hx hf ⟨hf.ne_zero, g, h.2, he.symm⟩ heval\n · refine aeval_ne_zero_of_dvdNotUnit_minpoly hx hg ⟨hg.ne_zero, f, h.1, ?_⟩ heval\n rw [mul_comm, he]\n\nend IsDomain\n\nend CommRing\n\n","line":273,"end_line":287,"proof_status":"formalized"} |
| {"id":"lem:gen_dvd_by_frak_p","label":"gen_dvd_by_frak_p","background_color":"#a3d6ff","border_color":"blue","kind":"ellipse","dep_graph_url":"https://leanprover-community.github.io/flt-regular/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) be a regular odd prime, \\(x,y,z,\\epsilon \\in \\mathbb {Z}[\\zeta _p]\\), \\(\\epsilon \\) a unit, and \\(n \\in \\mathbb {Z}_{\\geq 1}\\). Assume \\(x,y,z\\) are coprime to \\((1-\\zeta _p)\\) and that \\(x^p+y^p+\\epsilon (1-\\zeta _p)^{pn}z^p=0\\). Then each of \\((x+\\zeta _p^i y)\\) is divisible by \\((1-\\zeta _p)\\) and there is a unique \\(i_0\\) such that \\((x+\\zeta _p^{i_0})\\) is divisible by \\((1-\\zeta _p)^2\\).","doc_url":null,"latex_link":"sect0004.html#lem:gen_dvd_by_frak_p","lean_link":null,"lean_decl_name":"gen_dvd_by_frak_p","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
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