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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 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.CountableFieldEmbedding
namespace OAI
section
noncomputable section
open Set Function Cardinal
open scoped Classical
namespace CriticalPercolation.Arithmetic
lemma exists_transcendental_complex_norm_gt_one : ∃ z : ℂ, Transcendental ℚ z ∧ 1<‖z‖ := by
have hex : ∃ y : ℝ, Transcendental ℚ y := by
by_contra hn
push Not at hn
let : Algebra.IsAlgebraic ℚ ℝ := ⟨fun x => Classical.not_not.mp (hn x)⟩
have hh := Algebra.IsAlgebraic.cardinalMk_le_max ℚ ℝ
have hq : #(ℚ)≤ℵ₀ := mk_le_aleph0
rw [max_eq_right hq] at hh
exact (aleph0_lt_mk (α:=ℝ)).not_ge hh
obtain ⟨y,hy⟩ := hex
obtain ⟨n,hn⟩ := exists_nat_gt (‖y‖+2)
have hpos : 1<y+(n:ℝ) := by
have hab : -‖y‖≤y := by simpa only [Real.norm_eq_abs] using neg_abs_le y
linarith
have ht : Transcendental ℚ (y+(n:ℝ)) := by
intro halg
exact hy (by simpa only [add_sub_cancel_right] using halg.sub (isAlgebraic_natCast (R:=ℚ) n))
refine ⟨(y+(n:ℝ):ℝ),?_,?_⟩
· exact (transcendental_algebraMap_iff (R:=ℚ) (S:=ℝ) (A:=ℂ) (algebraMap ℝ ℂ).injective).mpr ht
· simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_pos (lt_trans zero_lt_one hpos)] using hpos
lemma exists_embedding_sending_transcendental {k K F : Type} [Field k] [Field K] [Field F]
[Algebra k K] [Algebra k F] [Countable k] [Countable K] [IsAlgClosed F] [Uncountable F]
(x : K) (hx : Transcendental k x) (y : F) (hy : Transcendental k y) :
∃ f : K →+* F, f x=y := by
let v : Unit → K := fun _ => x
let w : Unit → F := fun _ => y
have hv : AlgebraicIndependent k v := algebraicIndependent_unique_type_iff.mpr hx
have hw : AlgebraicIndependent k w := algebraicIndependent_unique_type_iff.mpr hy
let A := IntermediateField.adjoin k (Set.range v)
let φ : A →+* F := (IsFractionRing.lift
(algebraicIndependent_iff_injective_aeval.mp hw)).comp hv.reprField.toRingHom
let a : A := ⟨x,IntermediateField.subset_adjoin k (Set.range v) ⟨(),rfl⟩⟩
have hφ : φ a=y := by
let q : FractionRing (MvPolynomial Unit k) := algebraMap (MvPolynomial Unit k) _ (MvPolynomial.X ())
have ha : hv.aevalEquivField q=a := by
apply Subtype.ext
rw [hv.aevalEquivField_algebraMap_apply_coe]
simp only [MvPolynomial.aeval_X]
rfl
rw [←ha]
change IsFractionRing.lift (algebraicIndependent_iff_injective_aeval.mp hw)
(hv.aevalEquivField.symm (hv.aevalEquivField q))=y
rw [hv.aevalEquivField.symm_apply_apply]
simp only [q,IsFractionRing.lift_algebraMap]
exact MvPolynomial.aeval_X w ()
obtain ⟨f,hf⟩ := exists_extension_of_countable (k:=A) (K:=K) φ
refine ⟨f,?_⟩
exact (RingHom.congr_fun hf a).trans hφ
lemma exists_complex_embedding_norm_gt_one {K : Type} [Field K] [CharZero K] [Countable K]
(x : K) (hx : Transcendental ℚ x) : ∃ f : K →+* ℂ, 1<‖f x‖ := by
let : Uncountable ℂ := uncountable_complete_normed_field ℂ
obtain ⟨y,hy,hyNorm⟩ := exists_transcendental_complex_norm_gt_one
obtain ⟨f,hf⟩ := exists_embedding_sending_transcendental x hx y hy
exact ⟨f,hf ▸ hyNorm⟩
end CriticalPercolation.Arithmetic
end
end
end OAI
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