| import OAI.LinearAlgebra.CirculantHadamard.LocalDVR |
| import Mathlib.RingTheory.Localization.AtPrime.Basic |
| import Mathlib.RingTheory.Localization.Submodule |
| import Mathlib.RingTheory.Int.Basic |
| import Mathlib.Algebra.CharP.Algebra |
| import Mathlib.Algebra.Ring.Int.Parity |
|
|
| namespace OAI |
|
|
| /-! |
| # The integers localized at the prime two |
|
|
| The ring is the actual localization of `ℤ` at the complement of `(2)`. Its |
| maximal ideal is generated by the image of two, and its nonzero generator makes |
| it a discrete valuation ring. The characteristic-zero and odd-unit interfaces |
| refer to this concrete localization, without assuming any valuation facts. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace CirculantHadamard.IntegerTwoLocal |
|
|
| / |
| def ideal : Ideal ℤ := Ideal.span {(2 : ℤ)} |
| |
| instance ideal_isPrime : ideal.IsPrime := by |
| apply (Ideal.span_singleton_prime (by norm_num : (2 : ℤ) ≠ 0)).mpr |
| exact Nat.prime_iff_prime_int.mp Nat.prime_two |
| |
| / |
| abbrev Ring := Localization.AtPrime ideal |
| |
| / |
| theorem isDomain : IsDomain Ring := inferInstance |
| |
| / |
| theorem isLocalRing : IsLocalRing Ring := inferInstance |
| |
| / |
| theorem algebraMap_injective : Function.Injective (algebraMap ℤ Ring) := |
| IsLocalization.injective Ring ideal.primeCompl_le_nonZeroDivisors |
| |
| instance charZero : CharZero Ring := |
| charZero_of_injective_algebraMap algebraMap_injective |
| |
| / |
| instance isNoetherianRing : IsNoetherianRing Ring := |
| IsLocalization.isNoetherianRing ideal.primeCompl Ring |
| (inferInstance : IsNoetherianRing ℤ) |
| |
| / |
| theorem maximalIdeal_eq_span : |
| IsLocalRing.maximalIdeal Ring = Ideal.span {(2 : Ring)} := by |
| calc |
| IsLocalRing.maximalIdeal Ring = Ideal.map (algebraMap ℤ Ring) ideal := |
| (Localization.AtPrime.map_eq_maximalIdeal (I := ideal)).symm |
| _ = Ideal.span {(2 : Ring)} := by |
| simp only [ideal, Ideal.map_span, Set.image_singleton, map_ofNat] |
| |
| theorem two_ne_zero : (2 : Ring) ≠ 0 := by norm_num |
| |
| theorem two_not_isUnit : ¬ IsUnit (2 : Ring) := by |
| rw [← mem_nonunits_iff, ← IsLocalRing.mem_maximalIdeal, maximalIdeal_eq_span] |
| exact Ideal.mem_span_singleton_self _ |
| |
| / |
| (two); this conclusion is obtained from its proved principal maximal ideal. -/ |
| instance isDiscreteValuationRing : IsDiscreteValuationRing Ring := |
| LocalDVR.isDiscreteValuationRing_of_maximalIdeal_eq_span two_ne_zero |
| maximalIdeal_eq_span |
| |
| @[simp] theorem mem_ideal_iff_even (z : ℤ) : z ∈ ideal ↔ Even z := by |
| simp only [ideal, Ideal.mem_span_singleton, even_iff_two_dvd] |
| |
| / |
| theorem isUnit_algebraMap_iff_odd (z : ℤ) : |
| IsUnit (algebraMap ℤ Ring z) ↔ Odd z := by |
| rw [IsLocalization.AtPrime.isUnit_to_map_iff Ring ideal z] |
| change z ∉ ideal ↔ Odd z |
| rw [mem_ideal_iff_even, Int.not_even_iff_odd] |
| |
| theorem isUnit_intCast_of_odd (z : ℤ) (hz : Odd z) : IsUnit (z : Ring) := by |
| simpa using (isUnit_algebraMap_iff_odd z).mpr hz |
| |
| theorem isUnit_natCast_of_odd (u : ℕ) (hu : Odd u) : IsUnit (u : Ring) := by |
| simpa only [Int.cast_natCast] using |
| isUnit_intCast_of_odd (u : ℤ) ((Int.odd_coe_nat u).mpr hu) |
| |
| end CirculantHadamard.IntegerTwoLocal |
| |
| end |
| |
| end OAI |
| |
