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2c7d975 | 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 | import Mathlib
namespace OAI
noncomputable section
universe u
namespace Problem335
/-- Every countable characteristic-zero field admits an embedding into the complex numbers.
This applies to the subfield generated by the finitely many coefficients of a circuit. -/
theorem countable_field_embeds_complex (K : Type u) [Field K] [CharZero K] [Countable K] :
Nonempty (K →+* ℂ) := by
classical
obtain ⟨s, hs⟩ := exists_isTranscendenceBasis ℚ K
obtain ⟨t, ht⟩ := exists_isTranscendenceBasis ℚ ℂ
have htcard : Cardinal.mk ℂ = Cardinal.mk t :=
IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt'
(fun x : t => (x : ℂ)) ht Cardinal.mk_le_aleph0
(by simpa using Cardinal.aleph0_lt_continuum)
have : Infinite t := Cardinal.infinite_iff.mpr (by
rw [← htcard]
simpa using Cardinal.aleph0_lt_continuum.le)
let e : s ↪ t := (Classical.choice (nonempty_embedding_nat s)).trans
(Infinite.natEmbedding t)
let v : s → ℂ := fun x => (e x : ℂ)
have hv : AlgebraicIndependent ℚ v := ht.1.comp e e.injective
let A := Algebra.adjoin ℚ (Set.range (fun x : s => (x : K)))
let f : A →+* ℂ := (MvPolynomial.aeval v).toRingHom.comp hs.1.aevalEquiv.symm.toRingHom
have hf : Function.Injective f :=
(algebraicIndependent_iff_injective_aeval.mp hv).comp hs.1.aevalEquiv.symm.injective
let : Algebra A ℂ := f.toAlgebra
have : Module.IsTorsionFree A ℂ :=
(Module.isTorsionFree_iff_algebraMap_injective).mpr hf
have : Algebra.IsAlgebraic A K := hs.isAlgebraic
exact ⟨(IsAlgClosed.lift (R := A) (S := K) (M := ℂ)).toRingHom⟩
end Problem335
end
end OAI
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