openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /CountableFieldEmbedding.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Cardinal | |
| open scoped Classical | |
| namespace CriticalPercolation.Arithmetic | |
| lemma exists_algHom_of_countable {k K F : Type} [Field k] [Field K] [Field F] | |
| [Algebra k K] [Algebra k F] [Countable k] [Countable K] [IsAlgClosed F] [Uncountable F] : | |
| Nonempty (K →ₐ[k] F) := by | |
| obtain ⟨S,hS⟩ := exists_isTranscendenceBasis k K | |
| obtain ⟨T,hT⟩ := exists_isTranscendenceBasis k F | |
| have hcard : #F=#T := IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt' | |
| _ hT mk_le_aleph0 aleph0_lt_mk | |
| have hle : #S≤#T := by | |
| rw [←hcard] | |
| exact mk_le_aleph0.trans aleph0_lt_mk.le | |
| obtain ⟨e⟩ := (Cardinal.le_def S T).mp hle | |
| let y : S → F := fun s => (e s: F) | |
| have hy : AlgebraicIndependent k y := hT.1.comp e e.injective | |
| let A := IntermediateField.adjoin k (Set.range (Subtype.val : S → K)) | |
| let φ : A →ₐ[k] F := (IsFractionRing.liftAlgHom | |
| (algebraicIndependent_iff_injective_aeval.mp hy)).comp hS.1.reprField | |
| let : Algebra A F := φ.toRingHom.toAlgebra | |
| let : IsScalarTower k A F := IsScalarTower.of_algebraMap_eq (fun x => (φ.commutes x).symm) | |
| let : Algebra.IsAlgebraic A K := hS.isAlgebraic_field | |
| exact ⟨(IsAlgClosed.lift (R:=A) (S:=K) (M:=F)).restrictScalars k⟩ | |
| lemma exists_extension_of_countable {k K F : Type} [Field k] [Field K] [Field F] | |
| [Algebra k K] [Countable k] [Countable K] [IsAlgClosed F] [Uncountable F] | |
| (f : k →+* F) : ∃ g : K →+* F, g.comp (algebraMap k K)=f := by | |
| let : Algebra k F := f.toAlgebra | |
| obtain ⟨g⟩ := exists_algHom_of_countable (k:=k) (K:=K) (F:=F) | |
| exact ⟨g.toRingHom,g.comp_algebraMap⟩ | |
| lemma uncountable_complete_normed_field (F : Type*) [NontriviallyNormedField F] [CompleteSpace F] : | |
| Uncountable F := by | |
| obtain ⟨f,_,_,hf⟩ := (PerfectSpace.univ_perfect (α:=F)).exists_nat_bool_injection Set.univ_nonempty | |
| let : Uncountable (ℕ → Bool) := Cardinal.aleph0_lt_mk_iff.mp (by | |
| simpa only [Cardinal.mk_arrow, Cardinal.mk_bool, Cardinal.mk_nat, Cardinal.lift_id, Cardinal.two_power_aleph0] | |
| using Cardinal.aleph0_lt_continuum) | |
| exact hf.uncountable | |
| end CriticalPercolation.Arithmetic | |
| end | |
| end | |
| end OAI | |