File size: 2,293 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 | import Mathlib.RingTheory.Localization.AtPrime.Basic
import Mathlib.Algebra.Prime.Lemmas
import Mathlib.Data.Nat.Prime.Int
namespace OAI
universe uA
/-!
# Reflecting prime-power divisibility from the localized integers
A denominator inverted in the localization at `(p)` is not divisible by `p`.
Clearing that denominator and using primality therefore reflects divisibility
by every power of `p` back to the integers. The argument includes the zeroth
power and the zero numerator, without any nonzero numerator hypothesis.
-/
namespace CirculantHadamard
/-- Divisibility by a rational prime power in its actual integer localization
implies the same divisibility in the integers. -/
theorem integer_prime_pow_dvd_of_localization
{A : Type uA} [CommRing A] [Algebra ℤ A]
{P : Ideal ℤ} [P.IsPrime] [IsLocalization.AtPrime A P]
{p : ℕ} (hp : p.Prime) (hP : P = Ideal.span ({(p : ℤ)} : Set ℤ))
(t : ℕ) (x : ℤ) (hx : (p : A) ^ t ∣ algebraMap ℤ A x) :
(p : ℤ) ^ t ∣ x := by
obtain ⟨y, hy⟩ := hx
obtain ⟨⟨a, s⟩, hs⟩ := IsLocalization.surj P.primeCompl y
have hinj : Function.Injective (algebraMap ℤ A) :=
IsLocalization.injective A P.primeCompl_le_nonZeroDivisors
have hclear : x * (s : ℤ) = (p : ℤ) ^ t * a := by
apply hinj
rw [map_mul, map_mul, map_pow, map_natCast, hy, mul_assoc, hs]
have hsprime : ¬ (p : ℤ) ∣ (s : ℤ) := by
intro hdiv
have hmem : (s : ℤ) ∈ Ideal.span ({(p : ℤ)} : Set ℤ) :=
Ideal.mem_span_singleton.mpr hdiv
exact s.property (hP.symm ▸ hmem)
exact (Nat.prime_iff_prime_int.mp hp).pow_dvd_of_dvd_mul_right t hsprime
⟨a, hclear⟩
/-- Prime-power divisibility of integer coefficients is unchanged by
localization at that prime. -/
theorem integer_prime_pow_dvd_localization_iff
{A : Type uA} [CommRing A] [Algebra ℤ A]
{P : Ideal ℤ} [P.IsPrime] [IsLocalization.AtPrime A P]
{p : ℕ} (hp : p.Prime) (hP : P = Ideal.span ({(p : ℤ)} : Set ℤ))
(t : ℕ) (x : ℤ) :
(p : A) ^ t ∣ algebraMap ℤ A x ↔ (p : ℤ) ^ t ∣ x := by
constructor
· exact integer_prime_pow_dvd_of_localization hp hP t x
· intro hx
simpa only [map_pow, map_natCast] using map_dvd (algebraMap ℤ A) hx
end CirculantHadamard
end OAI
|