File size: 3,792 Bytes
cdec422 | 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 | import OAI.LinearAlgebra.CirculantHadamard.LocalizationMaps
import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
import Mathlib.RingTheory.LocalRing.ResidueField.Basic
import Mathlib.Algebra.CharP.Two
import Mathlib.Tactic.Ring
import Mathlib.Tactic.NormNum
namespace OAI
universe uB uO uk uF
/-!
# The characteristic-two local ring for the binary contradiction
The maximal ideal is constructed from an actual finite characteristic-zero
integer algebra. Unit denominators follow from the odd norm identity, not
from an assumed extension of complex conjugation to a localization.
-/
namespace CirculantHadamard.BinaryLocalization
/-- A finite characteristic-zero integer algebra has a maximal ideal over two. -/
theorem exists_maximal_two_mem (B : Type uB) [CommRing B] [CharZero B]
[Module.Finite ℤ B] :
∃ m : Ideal B, m.IsMaximal ∧ (2 : B) ∈ m := by
let : Algebra.IsIntegral ℤ B := Algebra.IsIntegral.of_finite ℤ B
obtain ⟨m, hm, _, htwo⟩ :=
exists_maximal_ideal_over_nat_prime_of_charZero B Nat.prime_two
exact ⟨m, hm, htwo⟩
/-- The one residue field used by both first-order maps has characteristic two. -/
theorem residue_char_two {B : Type uB} [CommRing B] (m : Ideal B) [m.IsPrime]
(htwo : (2 : B) ∈ m) :
CharP (IsLocalRing.ResidueField (Localization.AtPrime m)) 2 := by
let O := Localization.AtPrime m
apply CharTwo.of_one_ne_zero_of_two_eq_zero one_ne_zero
have hmem : (2 : O) ∈ IsLocalRing.maximalIdeal O := by
simpa only [map_ofNat] using
(IsLocalization.AtPrime.to_map_mem_maximal_iff O m (2 : B)).mpr htwo
have hz := (IsLocalRing.residue_eq_zero_iff (2 : O)).mpr hmem
simpa only [map_ofNat] using hz
/-- A square root of minus one reduces to one in a field of characteristic two. -/
theorem map_i_eq_one {O : Type uO} {k : Type uk} [CommRing O] [Field k] [CharP k 2]
(φ : O →+* k) (i : O) (hi : i ^ 2 = -1) : φ i = 1 := by
have htwo : (2 : k) = 0 := CharTwo.two_eq_zero
have hsquare : (φ i) ^ 2 = -1 := by
simpa using congrArg φ hi
have hsq : (φ i - 1) ^ 2 = 0 := by
calc
(φ i - 1) ^ 2 = (φ i) ^ 2 - 2 * φ i + 1 := by ring
_ = 0 := by rw [hsquare, htwo]; ring
exact sub_eq_zero.mp (eq_zero_of_pow_eq_zero hsq)
/-- An odd scalar is one in the common characteristic-two residue field. -/
theorem map_odd_nat_eq_one {O : Type uO} {k : Type uk} [CommRing O] [Field k] [CharP k 2]
(φ : O →+* k) {u : ℕ} (hu : Odd u) : φ (u : O) = 1 := by
rw [map_natCast]
exact natCast_eq_one_of_odd_of_two_eq_zero hu CharTwo.two_eq_zero
/-- The odd norm supplies a unit without requiring the second factor to be a
conjugation operation on the local ring. -/
theorem isUnit_of_mul_eq_odd_square {O : Type uO} [CommRing O] [IsLocalRing O]
[CharP (IsLocalRing.ResidueField O) 2] {x y : O} {u : ℕ}
(hu : Odd u) (hxy : x * y = (u : O) ^ 2) : IsUnit x := by
apply (IsLocalRing.residue_ne_zero_iff_isUnit x).mp
intro hx
have h := congrArg (IsLocalRing.residue O) hxy
have hu' := map_odd_nat_eq_one (IsLocalRing.residue O) hu
have hbad : (0 : IsLocalRing.ResidueField O) = 1 := by
simpa only [map_mul, map_pow, hx, zero_mul, hu', one_pow] using h
exact zero_ne_one hbad
/-- A faithful ring embedding reflects the odd-power witness for an actual unit. -/
theorem odd_power_of_map {O : Type uO} {F : Type uF} [CommRing O] [CommRing F]
(f : O →+* F) (hf : Function.Injective f) (u : Oˣ)
(h : ∃ m : ℕ, 0 < m ∧ Odd m ∧ (f (u : O)) ^ m = 1) :
∃ m : ℕ, 0 < m ∧ Odd m ∧ u ^ m = 1 := by
obtain ⟨m, hm, hodd, hpow⟩ := h
refine ⟨m, hm, hodd, ?_⟩
apply Units.ext
apply hf
simpa only [Units.val_pow_eq_pow_val, Units.val_one, map_pow, map_one] using hpow
end CirculantHadamard.BinaryLocalization
end OAI
|