| 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 |
|
|
| / |
| 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⟩ |
|
|
| / |
| 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 |
|
|
