Neg one not Mem powers q
ArkLib.Lattices.CyclotomicModulus.neg_one_notMem_powers_q
Plain-language statement
(Phase 1) −1 ∉ ⟨q⟩ in (Z/2^{α+1})ˣ, stated as: no power of q equals −1. For q ≡ 5 (mod 8), every power of q is ≡ 1 (mod 4) (since q ≡ 1 (mod 4)), while −1 ≡ 3 (mod 4) (as α ≥ 1, so 4 ∣ 2^{α+1}); projecting to ZMod 4 separates them. This is the precise sense in which q ≡ 5 (mod 8) forces σ_{-1} to swap the two CRT factors.
Exact Lean statement
theorem neg_one_notMem_powers_q (hq5 : q % 8 = 5) {α : ℕ} (hα : 1 ≤ α) :
∀ n : ℕ, (q : ZMod (2 ^ (α + 1))) ^ n ≠ -1Formal artifact
Lean source
theorem neg_one_notMem_powers_q (hq5 : q % 8 = 5) {α : ℕ} (hα : 1 ≤ α) : ∀ n : ℕ, (q : ZMod (2 ^ (α + 1))) ^ n ≠ -1 := by intro n hn have hdvd : (4 : ℕ) ∣ 2 ^ (α + 1) := by rw [show (4 : ℕ) = 2 ^ 2 from rfl]; exact pow_dvd_pow 2 (by omega) have hq4 : (q : ZMod 4) = 1 := by have h := (ZMod.natCast_eq_natCast_iff q 1 4).mpr (by rw [Nat.ModEq]; omega) simpa using h have hcast := congrArg (ZMod.castHom hdvd (ZMod 4)) hn rw [map_pow, map_neg, map_one, map_natCast, hq4, one_pow] at hcast exact absurd hcast (by decide)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/Factorization.lean:74-84
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Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.