openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /AllPlaceKronecker.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.ArchimedeanDetection | |
| import OAI.Probability.CriticalPercolation.LinearGroups.PadicDetection | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Cardinal Polynomial | |
| open scoped Classical | |
| namespace CriticalPercolation.Arithmetic | |
| lemma algebraic_of_all_complex_embeddings_bounded {K : Type} [Field K] [CharZero K] | |
| [Countable K] (x : K) (h : ∀ f : K →+* ℂ, ‖f x‖≤1) : IsAlgebraic ℚ x := by | |
| by_contra hn | |
| obtain ⟨f,hf⟩ := exists_complex_embedding_norm_gt_one x hn | |
| exact hf.not_ge (h f) | |
| lemma uncountable_padic_closure (p : ℕ) [Fact p.Prime] : | |
| Uncountable (AlgebraicClosure (Padic p)) := by | |
| let : Uncountable (Padic p) := uncountable_complete_normed_field (Padic p) | |
| exact (algebraMap (Padic p) (AlgebraicClosure (Padic p))).injective.uncountable | |
| lemma root_of_unity_of_all_places {K : Type} [Field K] [CharZero K] [Countable K] | |
| (x : K) (hx : x≠0) | |
| (hc : ∀ f : K →+* ℂ, ‖f x‖≤1) | |
| (hp : ∀ (p : ℕ) (_ : Fact p.Prime) (f : K →+* AlgebraicClosure (Padic p)), | |
| spectralNorm (Padic p) (AlgebraicClosure (Padic p)) (f x)≤1) : | |
| ∃ n : ℕ, 0<n ∧ x^n=1 := by | |
| have halg : IsAlgebraic ℚ x := algebraic_of_all_complex_embeddings_bounded x hc | |
| let A := IntermediateField.adjoin ℚ ({x}:Set K) | |
| let : FiniteDimensional ℚ A := IntermediateField.adjoin.finiteDimensional halg.isIntegral | |
| let : NumberField A := ⟨⟩ | |
| let a : A := ⟨x,IntermediateField.subset_adjoin ℚ _ (Set.mem_singleton x)⟩ | |
| have hcA : ∀ f : A →+* ℂ, ‖f a‖≤1 := by | |
| intro f | |
| let : Uncountable ℂ := uncountable_complete_normed_field ℂ | |
| obtain ⟨g,hg⟩ := exists_extension_of_countable (K:=K) f | |
| have he : g x=f a := RingHom.congr_fun hg a | |
| exact he ▸ hc g | |
| have hi : IsIntegral ℤ a := by | |
| apply integral_of_all_padic_embeddings | |
| intro p hp' f | |
| let : Fact p.Prime := hp' | |
| let : Uncountable (AlgebraicClosure (Padic p)) := uncountable_padic_closure p | |
| obtain ⟨g,hg⟩ := exists_extension_of_countable (K:=K) f | |
| have he : g x=f a := RingHom.congr_fun hg a | |
| exact he ▸ hp p hp' g | |
| obtain ⟨n,hn,he⟩ := NumberField.Embeddings.pow_eq_one_of_norm_le_one A ℂ | |
| (x:=a) (fun hz => hx (congrArg Subtype.val hz)) hi hcA | |
| exact ⟨n,hn,congrArg Subtype.val he⟩ | |
| end CriticalPercolation.Arithmetic | |
| end | |
| end | |
| end OAI | |