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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 | 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
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