| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| universe u |
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| namespace Problem335 |
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|
| /-- 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⟩ |
|
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| end Problem335 |
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| end |
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| end OAI |
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|
