| {"id":"MulSemiringAction.monic_charpoly","label":"MulSemiringAction.monic_charpoly","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.charpoly"],"nlp_text":"\\(F_b\\) is monic.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.monic_charpoly","latex_link":"Frobenius_project.html#MulSemiringAction.monic_charpoly","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.monic_charpoly","lean_decl_name":"MulSemiringAction.monic_charpoly","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/Invariant/Basic.lean","lean4_code":"theorem monic_charpoly (b : B) : (charpoly G b).Monic :=\n monic_prod_of_monic _ _ (fun _ _ ↦ monic_X_sub_C _)\n\n","line":134,"end_line":136,"proof_status":"formalized"} |
| {"id":"Algebra.IsInvariant.isIntegral","label":"Algebra.IsInvariant.isIntegral","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.monic_charpoly","Algebra.IsInvariant.charpoly_mem_lifts"],"nlp_text":"\\(B/A\\) is integral.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.isIntegral","latex_link":"Frobenius_project.html#Algebra.IsInvariant.isIntegral","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.isIntegral","lean_decl_name":"Algebra.IsInvariant.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/NumberTheory/NumberField/Basic.lean","lean4_code":"theorem isIntegral (x : 𝓞 K) : IsIntegral ℤ x := by\n obtain ⟨P, hPm, hP⟩ := x.isIntegral_coe\n refine ⟨P, hPm, ?_⟩\n rwa [IsScalarTower.algebraMap_eq (S := 𝓞 K), ← Polynomial.hom_eval₂, coe_eq_zero_iff] at hP\n\n","line":238,"end_line":242,"proof_status":"formalized"} |
| {"id":"fixed_of_fixed2","label":"fixed_of_fixed2","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["Algebra.IsInvariant.isIntegral","IsAlgebraic.exists_smul_eq_mul","fixed_of_fixed1"],"nlp_text":"Let \\(x \\in L\\). Suppose that \\(x\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then \\(x\\) is fixed by the automorphism group \\(\\operatorname{Aut}(L/K)\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed2","lean_link":null,"lean_decl_name":"fixed_of_fixed2","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/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed2 (f : L ≃ₐ[K] L) (x : L)\n (hx : ∀ g : MulAction.stabilizer G Q, IsFractionRing.stabilizerHom G P Q K L g x = x) :\n f x = x := by\n obtain ⟨_⟩ := nonempty_fintype G\n have : P.IsPrime := Ideal.over_def Q P ▸ Ideal.IsPrime.under A Q\n have : Algebra.IsIntegral A B := Algebra.IsInvariant.isIntegral A B G\n obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := B ⧸ Q) x\n obtain ⟨b, a, ha, h⟩ := (Algebra.IsAlgebraic.isAlgebraic (R := A ⧸ P) y).exists_smul_eq_mul x hy\n replace ha : algebraMap (A ⧸ P) L a ≠ 0 := by\n rwa [Ne, algebraMap_apply (A ⧸ P) K L, algebraMap_eq_zero_iff, algebraMap_eq_zero_iff]\n replace hy : algebraMap (B ⧸ Q) L y ≠ 0 :=\n mt (algebraMap_eq_zero_iff (B ⧸ Q) L).mp (nonZeroDivisors.ne_zero hy)\n replace h : algebraMap (B ⧸ Q) L x / algebraMap (B ⧸ Q) L y =\n algebraMap (B ⧸ Q) L b / algebraMap (A ⧸ P) L a := by\n rw [mul_comm, Algebra.smul_def, mul_comm] at h\n rw [div_eq_div_iff hy ha, ← map_mul, ← h, map_mul, ← algebraMap_apply]\n simp only [h, map_div₀, algebraMap_apply (A ⧸ P) K L, AlgEquiv.commutes] at hx ⊢\n simp only [← algebraMap_apply, div_left_inj' ha] at hx ⊢\n exact fixed_of_fixed1 G P Q K L f b (fun g ↦ IsFractionRing.injective (B ⧸ Q) L\n ((IsFractionRing.fieldEquivOfAlgEquiv_algebraMap K L L\n (Ideal.Quotient.stabilizerHom Q P G g) b).symm.trans (hx g)))\n\n/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\n","line":342,"end_line":364,"proof_status":"formalized"} |
| {"id":"IsFractionRing.stabilizerHom_surjective","label":"IsFractionRing.stabilizerHom_surjective","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed2","FixedPoints.toAlgAut_surjective"],"nlp_text":"The map \\(g\\mapsto \\phi _g\\) from \\(D_Q\\) to \\(\\operatorname{Aut}(L/K)\\) defined above is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom_surjective","latex_link":"Frobenius_project.html#IsFractionRing.stabilizerHom_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom_surjective","lean_decl_name":"IsFractionRing.stabilizerHom_surjective","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/RingTheory/Invariant/Basic.lean","lean4_code":"/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\ntheorem IsFractionRing.stabilizerHom_surjective :\n Function.Surjective (stabilizerHom G P Q K L) := by\n let _ := MulSemiringAction.compHom L (stabilizerHom G P Q K L)\n intro f\n obtain ⟨g, hg⟩ := FixedPoints.toAlgAut_surjective (MulAction.stabilizer G Q) L\n (AlgEquiv.ofRingEquiv (f := f) (fun x ↦ fixed_of_fixed2 G P Q K L f x x.2))\n exact ⟨g, by rwa [AlgEquiv.ext_iff] at hg ⊢⟩","line":375,"end_line":382,"proof_status":"formalized"} |
| {"id":"Algebra.IsInvariant.charpoly_mem_lifts","label":"Algebra.IsInvariant.charpoly_mem_lifts","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["MulSemiringAction.charpoly"],"nlp_text":"\\(F_b\\) is the lift of some monic polynomial \\(M_b\\) in \\(A[X]\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.charpoly_mem_lifts","latex_link":"Frobenius_project.html#Algebra.IsInvariant.charpoly_mem_lifts","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Algebra.IsInvariant.charpoly_mem_lifts","lean_decl_name":"Algebra.IsInvariant.charpoly_mem_lifts","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/Invariant/Basic.lean","lean4_code":"theorem charpoly_mem_lifts [Fintype G] (b : B) :\n charpoly G b ∈ Polynomial.lifts (algebraMap A B) :=\n (charpoly G b).lifts_iff_coeff_lifts.mpr fun n ↦ isInvariant _ (smul_coeff_charpoly b n)\n\n","line":159,"end_line":162,"proof_status":"formalized"} |
| {"id":"fixed_of_fixed1_aux1","label":"fixed_of_fixed1_aux1","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["Algebra.IsInvariant.charpoly_mem_lifts"],"nlp_text":"There exist elements \\(a,b \\in B\\), with \\(a \\notin Q\\) and \\(a\\) in the image of \\(A\\) such that for all \\(g\\in G\\), If \\(g \\cdot Q = Q\\), then \\(g \\cdot b \\equiv a \\pmod{Q}\\). If \\(g \\cdot Q \\neq Q\\), then \\(g \\cdot b \\equiv 0 \\pmod{Q}\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1_aux1","lean_link":null,"lean_decl_name":"fixed_of_fixed1_aux1","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/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1_aux1 [DecidableEq (Ideal B)] :\n ∃ a b : B, (∀ g : G, g • a = a) ∧ a ∉ Q ∧\n ∀ g : G, algebraMap B (B ⧸ Q) (g • b) = algebraMap B (B ⧸ Q) (if g • Q = Q then a else 0) := by\n obtain ⟨_⟩ := nonempty_fintype G\n let P := Finset.inf {g : G | g • Q ≠ Q} (fun g ↦ g • Q)\n have h1 : ¬ P ≤ Q := by\n rw [Ideal.IsPrime.inf_le' inferInstance]\n rintro ⟨g, hg1, hg2⟩\n exact (Finset.mem_filter.mp hg1).2 (smul_eq_of_smul_le hg2)\n obtain ⟨b, hbP, hbQ⟩ := SetLike.not_le_iff_exists.mp h1\n replace hbP : ∀ g : G, g • Q ≠ Q → b ∈ g • Q :=\n fun g hg ↦ (Finset.inf_le (Finset.mem_filter.mpr ⟨Finset.mem_univ g, hg⟩) : P ≤ g • Q) hbP\n let f := MulSemiringAction.charpoly G b\n obtain ⟨q, hq, hq0⟩ :=\n (f.map (algebraMap B (B ⧸ Q))).exists_eq_pow_rootMultiplicity_mul_and_not_dvd\n (Polynomial.map_monic_ne_zero (MulSemiringAction.monic_charpoly G b)) 0\n rw [map_zero, sub_zero] at hq hq0\n let j := (f.map (algebraMap B (B ⧸ Q))).rootMultiplicity 0\n let k := q.natDegree\n let r := ∑ i ∈ Finset.range (k + 1), Polynomial.monomial i (f.coeff (i + j))\n have hr : r.map (algebraMap B (B ⧸ Q)) = q := by\n ext n\n rw [Polynomial.coeff_map, Polynomial.finset_sum_coeff]\n simp only [Polynomial.coeff_monomial, Finset.sum_ite_eq', Finset.mem_range_succ_iff]\n split_ifs with hn\n · rw [← Polynomial.coeff_map, hq, Polynomial.coeff_X_pow_mul]\n · rw [map_zero, eq_comm, Polynomial.coeff_eq_zero_of_natDegree_lt (lt_of_not_ge hn)]\n have hf : f.eval b = 0 := MulSemiringAction.eval_charpoly G b\n have hr : r.eval b ∈ Q := by\n rw [← Ideal.Quotient.eq_zero_iff_mem, ← Ideal.Quotient.algebraMap_eq] at hbQ ⊢\n replace hf := congrArg (algebraMap B (B ⧸ Q)) hf\n rw [← Polynomial.eval₂_at_apply, ← Polynomial.eval_map] at hf ⊢\n rwa [map_zero, hq, ← hr, Polynomial.eval_mul, Polynomial.eval_pow, Polynomial.eval_X,\n mul_eq_zero, or_iff_right (pow_ne_zero _ hbQ)] at hf\n let a := f.coeff j\n have ha : ∀ g : G, g • a = a := MulSemiringAction.smul_coeff_charpoly b j\n have hr' : ∀ g : G, g • Q ≠ Q → a - r.eval b ∈ g • Q := by\n intro g hg\n have hr : r = ∑ i ∈ Finset.range (k + 1), Polynomial.monomial i (f.coeff (i + j)) := rfl\n rw [← Ideal.neg_mem_iff, neg_sub, hr, Finset.sum_range_succ', Polynomial.eval_add,\n Polynomial.eval_monomial, zero_add, pow_zero, mul_one, add_sub_cancel_right]\n simp only [ ← Polynomial.monomial_mul_X]\n rw [← Finset.sum_mul, Polynomial.eval_mul_X]\n exact Ideal.mul_mem_left (g • Q) _ (hbP g hg)\n refine ⟨a, a - r.eval b, ha, ?_, fun h ↦ ?_⟩\n · rwa [← Ideal.Quotient.eq_zero_iff_mem, ← Ideal.Quotient.algebraMap_eq, ← Polynomial.coeff_map,\n ← zero_add j, hq, Polynomial.coeff_X_pow_mul, ← Polynomial.X_dvd_iff]\n · rw [← sub_eq_zero, ← map_sub, Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem,\n ← Ideal.smul_mem_pointwise_smul_iff (a := h⁻¹), smul_sub, inv_smul_smul]\n simp only [← eq_inv_smul_iff (g := h), eq_comm (a := Q)]\n","line":218,"end_line":267,"proof_status":"formalized"} |
| {"id":"fixed_of_fixed1_aux2","label":"fixed_of_fixed1_aux2","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed1_aux1"],"nlp_text":"Let \\(b_0 \\in B\\). Suppose that the image of \\(b_0\\) in the quotient \\(B/Q\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then there exist elements \\(a,b \\in B\\), with \\(a\\notin Q\\) and \\(a\\) in the image of \\(A\\) such that for all \\(g\\in G\\), If \\(g \\cdot Q = Q\\), then \\(g \\cdot b \\equiv ab_0 \\pmod{Q}\\). If \\(g \\cdot Q \\neq Q\\), then \\(g \\cdot b \\equiv 0 \\pmod{Q}\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1_aux2","lean_link":null,"lean_decl_name":"fixed_of_fixed1_aux2","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/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1_aux2 [DecidableEq (Ideal B)] (b₀ : B)\n (hx : ∀ g : G, g • Q = Q → algebraMap B (B ⧸ Q) (g • b₀) = algebraMap B (B ⧸ Q) b₀) :\n ∃ a b : B, (∀ g : G, g • a = a) ∧ a ∉ Q ∧\n (∀ g : G, algebraMap B (B ⧸ Q) (g • b) =\n algebraMap B (B ⧸ Q) (if g • Q = Q then a * b₀ else 0)) := by\n obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux1 G Q\n refine ⟨a, b * b₀, ha1, ha2, fun g ↦ ?_⟩\n rw [smul_mul', map_mul, hb]\n specialize hb g\n split_ifs with hg\n · rw [map_mul, hx g hg]\n · rw [map_zero, zero_mul]\n\n/-- A technical lemma for `fixed_of_fixed1`. -/\nprivate theorem fixed_of_fixed1_aux3 [NoZeroDivisors B] {b : B} {i j : ℕ} {p : Polynomial A}\n (h : p.map (algebraMap A B) = (X - C b) ^ i * X ^ j) (f : B ≃ₐ[A] B) (hi : i ≠ 0) :\n f b = b := by\n by_cases ha : b = 0\n · rw [ha, map_zero]\n have hf := congrArg (eval b) (congrArg (Polynomial.mapAlgHom f.toAlgHom) h)\n rw [coe_mapAlgHom, map_map, f.toAlgHom.comp_algebraMap, h] at hf\n simp_rw [Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_sub, map_X, map_C,\n eval_mul, eval_pow, eval_sub, eval_X, eval_C, sub_self, zero_pow hi, zero_mul,\n zero_eq_mul, or_iff_left (pow_ne_zero j ha), pow_eq_zero_iff hi, sub_eq_zero] at hf\n exact hf.symm\n\n/-- This theorem will be made redundant by `IsFractionRing.stabilizerHom_surjective`. -/\nprivate theorem fixed_of_fixed1 [NoZeroSMulDivisors (B ⧸ Q) L] (f : L ≃ₐ[K] L) (b : B ⧸ Q)\n (hx : ∀ g : MulAction.stabilizer G Q, Ideal.Quotient.stabilizerHom Q P G g b = b) :\n f (algebraMap (B ⧸ Q) L b) = (algebraMap (B ⧸ Q) L b) := by\n classical\n cases nonempty_fintype G\n obtain ⟨b₀, rfl⟩ := Ideal.Quotient.mk_surjective b\n rw [← Ideal.Quotient.algebraMap_eq]\n obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux2 G Q b₀ (fun g hg ↦ hx ⟨g, hg⟩)\n obtain ⟨M, key⟩ := (mem_lifts _).mp (Algebra.IsInvariant.charpoly_mem_lifts A B G b)\n replace key := congrArg (map (algebraMap B (B ⧸ Q))) key\n rw [map_map, ← algebraMap_eq, algebraMap_eq A (A ⧸ P) (B ⧸ Q),\n ← map_map, MulSemiringAction.charpoly, Polynomial.map_prod] at key\n have key₀ : ∀ g : G, (X - C (g • b)).map (algebraMap B (B ⧸ Q)) =\n if g • Q = Q then X - C (algebraMap B (B ⧸ Q) (a * b₀)) else X := by\n intro g\n rw [Polynomial.map_sub, map_X, map_C, hb]\n split_ifs\n · rfl\n · rw [map_zero, map_zero, sub_zero]\n simp only [key₀, Finset.prod_ite, Finset.prod_const] at key\n replace key := congrArg (map (algebraMap (B ⧸ Q) L)) key\n rw [map_map, ← algebraMap_eq, algebraMap_eq (A ⧸ P) K L,\n ← map_map, Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_sub,\n","line":274,"end_line":323,"proof_status":"formalized"} |
| {"id":"fixed_of_fixed1","label":"fixed_of_fixed1","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":["fixed_of_fixed1_aux2","IsFractionRing.stabilizerHom"],"nlp_text":"Let \\(b_0 \\in B/Q\\). Suppose that \\(b_0\\) is fixed by the stabilizer subgroup \\(D_Q\\). Then \\(b_0\\) is fixed by \\(\\operatorname{Aut}(L/K)\\).","doc_url":null,"latex_link":"Frobenius_project.html#fixed_of_fixed1","lean_link":null,"lean_decl_name":"fixed_of_fixed1","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/Invariant/Basic.lean","lean4_code":"private theorem fixed_of_fixed1 [NoZeroSMulDivisors (B ⧸ Q) L] (f : L ≃ₐ[K] L) (b : B ⧸ Q)\n (hx : ∀ g : MulAction.stabilizer G Q, Ideal.Quotient.stabilizerHom Q P G g b = b) :\n f (algebraMap (B ⧸ Q) L b) = (algebraMap (B ⧸ Q) L b) := by\n classical\n cases nonempty_fintype G\n obtain ⟨b₀, rfl⟩ := Ideal.Quotient.mk_surjective b\n rw [← Ideal.Quotient.algebraMap_eq]\n obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux2 G Q b₀ (fun g hg ↦ hx ⟨g, hg⟩)\n obtain ⟨M, key⟩ := (mem_lifts _).mp (Algebra.IsInvariant.charpoly_mem_lifts A B G b)\n replace key := congrArg (map (algebraMap B (B ⧸ Q))) key\n rw [map_map, ← algebraMap_eq, algebraMap_eq A (A ⧸ P) (B ⧸ Q),\n ← map_map, MulSemiringAction.charpoly, Polynomial.map_prod] at key\n have key₀ : ∀ g : G, (X - C (g • b)).map (algebraMap B (B ⧸ Q)) =\n if g • Q = Q then X - C (algebraMap B (B ⧸ Q) (a * b₀)) else X := by\n intro g\n rw [Polynomial.map_sub, map_X, map_C, hb]\n split_ifs\n · rfl\n · rw [map_zero, map_zero, sub_zero]\n simp only [key₀, Finset.prod_ite, Finset.prod_const] at key\n replace key := congrArg (map (algebraMap (B ⧸ Q) L)) key\n rw [map_map, ← algebraMap_eq, algebraMap_eq (A ⧸ P) K L,\n ← map_map, Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_sub,\n map_X, map_C] at key\n replace key := fixed_of_fixed1_aux3 key f (Finset.card_ne_zero_of_mem\n (Finset.mem_filter.mpr ⟨Finset.mem_univ 1, one_smul G Q⟩))\n simp only [map_mul] at key\n obtain ⟨a, rfl⟩ := Algebra.IsInvariant.isInvariant (A := A) a ha1\n rwa [← algebraMap_apply A B (B ⧸ Q), algebraMap_apply A (A ⧸ P) (B ⧸ Q),\n ← algebraMap_apply, algebraMap_apply (A ⧸ P) K L, f.commutes, mul_right_inj'] at key\n rwa [← algebraMap_apply, algebraMap_apply (A ⧸ P) (B ⧸ Q) L,\n ← algebraMap_apply A (A ⧸ P) (B ⧸ Q), algebraMap_apply A B (B ⧸ Q),\n Ne, algebraMap_eq_zero_iff, Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem]\n\nvariable [IsFractionRing (A ⧸ P) K] [IsFractionRing (B ⧸ Q) L]\n\n/-- If `Q` lies over `P`, then the stabilizer of `Q` acts on `Frac(B/Q)/Frac(A/P)`. -/\n","line":301,"end_line":337,"proof_status":"formalized"} |
| {"id":"MulSemiringAction.charpoly","label":"MulSemiringAction.charpoly","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"If \\(b\\in B\\) then define the characteristic polynomial \\(F_b(X) \\in B[X]\\) of \\(b\\) to be \\(\\prod _{g\\in G}(X-g\\cdot b)\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.charpoly","latex_link":"Frobenius_project.html#MulSemiringAction.charpoly","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/MulSemiringAction.charpoly","lean_decl_name":"MulSemiringAction.charpoly","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch8","source_path":"./FLT/Deformations/RepresentationTheory/GaloisRep.lean","lean4_code":"def GaloisRep.charFrob (ρ : GaloisRep K A M) : Polynomial A := (ρ.toLocal v Frobᵥ).charpoly\n\nomit [IsTopologicalRing A] in\n","line":316,"end_line":318,"proof_status":"formalized"} |
| {"id":"IsAlgebraic.exists_smul_eq_mul","label":"IsAlgebraic.exists_smul_eq_mul","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"If \\(R/S\\) is an algebraic extension of integral domains, then any fraction \\(a/b\\) with \\(a,b\\in R\\) can be written as \\(c/d\\) with \\(c\\in R\\) and \\(d\\in S\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsAlgebraic.exists_smul_eq_mul","latex_link":"Frobenius_project.html#IsAlgebraic.exists_smul_eq_mul","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsAlgebraic.exists_smul_eq_mul","lean_decl_name":"IsAlgebraic.exists_smul_eq_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/RingTheory/Algebraic/Basic.lean","lean4_code":"/-- A fraction `(a : S) / (b : S)` can be reduced to `(c : S) / (d : R)`,\nif `b` is algebraic over `R`. -/\ntheorem IsAlgebraic.exists_smul_eq_mul\n (a : S) {b : S} (hRb : IsAlgebraic R b) (hb : b ∈ S⁰) :\n ∃ᵉ (c : S) (d ≠ (0 : R)), d • a = b * c :=\n have ⟨r, hr, s, h⟩ := hRb.exists_nonzero_dvd hb\n ⟨s * a, r, hr, by rw [Algebra.smul_def, h, mul_assoc]⟩","line":557,"end_line":563,"proof_status":"formalized"} |
| {"id":"IsFractionRing.stabilizerHom","label":"IsFractionRing.stabilizerHom","background_color":"#B0ECA3","border_color":"darkgreen","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"Choose \\(g\\in D_Q\\). Then the action of \\(g\\) on \\(B\\) gives us an induced \\(A/P\\)-algebra automorphism of \\(B/Q\\) which extends to a \\(K\\)-algebra automorphism \\(\\phi (g)\\) of \\(L\\). This construction \\(g\\mapsto \\phi (g)\\) defines a group homomorphism from \\(D_Q\\) to \\(\\operatorname{Aut}(L/K)\\) (all the proofs implicit in the definition here are straightforward).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom","latex_link":"Frobenius_project.html#IsFractionRing.stabilizerHom","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/IsFractionRing.stabilizerHom","lean_decl_name":"IsFractionRing.stabilizerHom","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/Invariant/Basic.lean","lean4_code":"noncomputable def IsFractionRing.stabilizerHom : MulAction.stabilizer G Q →* (L ≃ₐ[K] L) :=\n MonoidHom.comp (IsFractionRing.fieldEquivOfAlgEquivHom K L) (Ideal.Quotient.stabilizerHom Q P G)\n\n/-- This theorem will be made redundant by `IsFractionRing.stabilizerHom_surjective`. -/\nprivate theorem fixed_of_fixed2 (f : L ≃ₐ[K] L) (x : L)\n (hx : ∀ g : MulAction.stabilizer G Q, IsFractionRing.stabilizerHom G P Q K L g x = x) :\n f x = x := by\n obtain ⟨_⟩ := nonempty_fintype G\n have : P.IsPrime := Ideal.over_def Q P ▸ Ideal.IsPrime.under A Q\n have : Algebra.IsIntegral A B := Algebra.IsInvariant.isIntegral A B G\n obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := B ⧸ Q) x\n obtain ⟨b, a, ha, h⟩ := (Algebra.IsAlgebraic.isAlgebraic (R := A ⧸ P) y).exists_smul_eq_mul x hy\n replace ha : algebraMap (A ⧸ P) L a ≠ 0 := by\n rwa [Ne, algebraMap_apply (A ⧸ P) K L, algebraMap_eq_zero_iff, algebraMap_eq_zero_iff]\n replace hy : algebraMap (B ⧸ Q) L y ≠ 0 :=\n mt (algebraMap_eq_zero_iff (B ⧸ Q) L).mp (nonZeroDivisors.ne_zero hy)\n replace h : algebraMap (B ⧸ Q) L x / algebraMap (B ⧸ Q) L y =\n algebraMap (B ⧸ Q) L b / algebraMap (A ⧸ P) L a := by\n rw [mul_comm, Algebra.smul_def, mul_comm] at h\n rw [div_eq_div_iff hy ha, ← map_mul, ← h, map_mul, ← algebraMap_apply]\n simp only [h, map_div₀, algebraMap_apply (A ⧸ P) K L, AlgEquiv.commutes] at hx ⊢\n simp only [← algebraMap_apply, div_left_inj' ha] at hx ⊢\n exact fixed_of_fixed1 G P Q K L f b (fun g ↦ IsFractionRing.injective (B ⧸ Q) L\n ((IsFractionRing.fieldEquivOfAlgEquiv_algebraMap K L L\n (Ideal.Quotient.stabilizerHom Q P G g) b).symm.trans (hx g)))\n\n/-- The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`. -/\n","line":338,"end_line":364,"proof_status":"formalized"} |
| {"id":"FixedPoints.toAlgAut_surjective","label":"FixedPoints.toAlgAut_surjective","background_color":"#1CAC78","border_color":"darkgreen","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_8.html","dependencies":[],"nlp_text":"Let \\(H\\) be a finite group acting on a field \\(F\\) by field automorphisms. Then the map \\(H \\to \\operatorname{Aut}(F/F^H)\\) is surjective.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FixedPoints.toAlgAut_surjective","latex_link":"Frobenius_project.html#FixedPoints.toAlgAut_surjective","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FixedPoints.toAlgAut_surjective","lean_decl_name":"FixedPoints.toAlgAut_surjective","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/Fixed.lean","lean4_code":"theorem toAlgAut_surjective [Finite G] :\n Function.Surjective (MulSemiringAction.toAlgAut G (FixedPoints.subfield G F) F) := by\n let f : G →* F ≃ₐ[FixedPoints.subfield G F] F :=\n MulSemiringAction.toAlgAut G (FixedPoints.subfield G F) F\n let Q := G ⧸ f.ker\n let _ : MulSemiringAction Q F := MulSemiringAction.compHom _ (QuotientGroup.kerLift f)\n have : FaithfulSMul Q F := ⟨by\n intro q₁ q₂\n refine Quotient.inductionOn₂' q₁ q₂ (fun g₁ g₂ h ↦ QuotientGroup.eq.mpr ?_)\n rwa [MonoidHom.mem_ker, map_mul, map_inv, inv_mul_eq_one, AlgEquiv.ext_iff]⟩\n intro f\n obtain ⟨q, hq⟩ := (toAlgAut_bijective Q F).surjective\n (AlgEquiv.ofRingEquiv (f := f) (fun ⟨x, hx⟩ ↦ f.commutes' ⟨x, fun g ↦ hx g⟩))\n revert hq\n refine QuotientGroup.induction_on q (fun g hg ↦ ⟨g, ?_⟩)\n rwa [AlgEquiv.ext_iff] at hg ⊢\n\n","line":360,"end_line":376,"proof_status":"formalized"} |
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