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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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function Cardinal Polynomial
open scoped Classical
namespace CriticalPercolation.Arithmetic
lemma rat_exists_prime_norm_gt_one (q : ℚ) (hq : q.den≠1) :
∃ p : ℕ, p.Prime ∧ 1<padicNorm p q := by
obtain ⟨p,hp,hpd⟩ := Nat.exists_prime_and_dvd hq
let : Fact p.Prime := ⟨hp⟩
have hpn : ¬p∣q.num.natAbs := by
intro hd
have hh := Nat.dvd_gcd hd hpd
rw [q.reduced] at hh
exact hp.not_dvd_one hh
have hnum : padicNorm p (q.num:ℚ)=1 :=
(padicNorm.int_eq_one_iff _).mpr (by simpa only [Int.natCast_dvd] using hpn)
have hden : padicNorm p (q.den:ℚ)<1 := (padicNorm.nat_lt_one_iff _).mpr hpd
have hpos : 0<padicNorm p (q.den:ℚ) := lt_of_le_of_ne (padicNorm.nonneg _) (Ne.symm
(padicNorm.nonzero (by exact_mod_cast q.den_nz)))
refine ⟨p,hp,?_⟩
conv_rhs => rw [←Rat.num_div_den q]
rw [padicNorm.div,hnum]
exact (one_lt_div hpos).mpr (by simpa only [one_mul] using hden)
lemma rat_integer_of_all_padic_norm_le_one (q : ℚ)
(h : ∀ p : ℕ, p.Prime → padicNorm p q≤1) : ∃ z : ℤ, (z:ℚ)=q := by
have hd : q.den=1 := by
by_contra hn
obtain ⟨p,hp,hpq⟩ := rat_exists_prime_norm_gt_one q hn
exact hpq.not_ge (h p hp)
exact ⟨q.num,by simpa only [hd,Nat.cast_one,div_one] using (Rat.num_div_den q)⟩
lemma coeff_norm_le_one_of_roots {F : Type*} [NormedField F]
(hna : IsNonarchimedean (norm : F → ℝ)) (P : F[X]) (hm : P.Monic) (hs : P.Splits)
(hr : ∀ z∈P.roots, ‖z‖≤1) (i : ℕ) : ‖P.coeff i‖≤1 := by
let R : Subring F := {
carrier := {z | ‖z‖≤1}
zero_mem' := by simp
one_mem' := by simp
add_mem' := fun hx hy => (hna _ _).trans (max_le hx hy)
mul_mem' := fun hx hy => by
change ‖_ * _‖≤1
rw [norm_mul]
exact (mul_le_mul hx hy (norm_nonneg _) zero_le_one).trans_eq (one_mul _)
neg_mem' := fun hx => by simpa only [Set.mem_ofPred_eq,norm_neg] using hx }
have hP := hs.mem_lift_of_roots_mem_range hm R.subtype (fun z hz => ⟨⟨z,hr z hz⟩,rfl⟩)
obtain ⟨Q,hQ⟩ := hP
change Q.map R.subtype=P at hQ
rw [←hQ,coeff_map]
exact (Q.coeff i).property
lemma integral_of_all_padic_embeddings {A : Type} [Field A] [NumberField A] (x : A)
(h : ∀ (p : ℕ) (_ : Fact p.Prime) (f : A →+* AlgebraicClosure (Padic p)),
spectralNorm (Padic p) (AlgebraicClosure (Padic p)) (f x)≤1) : IsIntegral ℤ x := by
have hcoeff : ∀ i, ∃ z : ℤ, (z:ℚ)=(minpoly ℚ x).coeff i := by
intro i
apply rat_integer_of_all_padic_norm_le_one
intro p hp
let : Fact p.Prime := ⟨hp⟩
let F := AlgebraicClosure (Padic p)
let : NontriviallyNormedField F := spectralNorm.nontriviallyNormedField (Padic p) F
let : NormedAlgebra (Padic p) F := spectralNorm.normedAlgebra (Padic p) F
have hh : ‖((minpoly ℚ x).map (algebraMap ℚ F)).coeff i‖≤1 := by
apply coeff_norm_le_one_of_roots (isNonarchimedean_spectralNorm (K:=Padic p) (L:=F))
_ ((minpoly.monic (Algebra.IsSeparable.isIntegral ℚ x)).map _) (IsAlgClosed.splits _)
intro z hz
have hroot : z∈(minpoly ℚ x).rootSet F := by
exact Multiset.mem_toFinset.mpr hz
obtain ⟨f,hf⟩ := (NumberField.Embeddings.range_eval_eq_rootSet_minpoly A F x).symm.subset hroot
rw [←hf]
exact h p inferInstance f
have he : ‖algebraMap ℚ F ((minpoly ℚ x).coeff i)‖ =
(padicNorm p ((minpoly ℚ x).coeff i):ℝ) := by
rw [IsScalarTower.algebraMap_apply ℚ (Padic p) F,norm_algebraMap']
exact Padic.eq_padicNorm _
rw [coeff_map,he] at hh
exact_mod_cast hh
have hl : minpoly ℚ x ∈ Polynomial.lifts (Int.castRingHom ℚ) := by
apply (Polynomial.lifts_iff_coeff_lifts _).mpr
intro i
exact hcoeff i
obtain ⟨P,hP⟩ := hl
change P.map (Int.castRingHom ℚ)=minpoly ℚ x at hP
refine ⟨P,?_,?_⟩
· apply Polynomial.monic_of_injective (f:=Int.castRingHom ℚ) (Int.cast_injective (α:=ℚ))
simpa only [hP] using minpoly.monic (Algebra.IsSeparable.isIntegral ℚ x)
· change Polynomial.aeval x P=0
rw [←Polynomial.aeval_map_algebraMap ℚ]
change Polynomial.aeval x (P.map (Int.castRingHom ℚ))=0
rw [hP]
exact minpoly.aeval ℚ x
end CriticalPercolation.Arithmetic
end
end
end OAI
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