Cyclotomic card normalized Factors
ArkLib.Lattices.CyclotomicModulus.cyclotomic_card_normalizedFactors
Plain-language statement
(Phase 2) Φ_{2^{α+1}} over Z_q has exactly two distinct monic irreducible factors, each of degree 2^{α-1} = d/2, for q ≡ 5 (mod 8). From Mathlib's normalizedFactors_cyclotomic_card (count = φ(2^{α+1}) / ord = 2^α / 2^{α-1} = 2) and natDegree_of_mem_normalizedFactors_cyclotomic (degree = ord = 2^{α-1}), using orderOf_q_eq.
Exact Lean statement
theorem cyclotomic_card_normalizedFactors (hq5 : q % 8 = 5) {α : ℕ} (hα : 1 ≤ α) :
(UniqueFactorizationMonoid.normalizedFactors
(cyclotomic (2 ^ (α + 1)) (ZMod q))).toFinset.card = 2Formal artifact
Lean source
theorem cyclotomic_card_normalizedFactors (hq5 : q % 8 = 5) {α : ℕ} (hα : 1 ≤ α) : (UniqueFactorizationMonoid.normalizedFactors (cyclotomic (2 ^ (α + 1)) (ZMod q))).toFinset.card = 2 := by have hcard : Fintype.card (ZMod q) = q ^ 1 := by rw [pow_one]; exact ZMod.card q have hcop : Nat.Coprime q (2 ^ (α + 1)) := Nat.Coprime.pow_right _ ((Nat.coprime_primes (Fact.out : q.Prime) Nat.prime_two).mpr (by omega)) rw [Polynomial.normalizedFactors_cyclotomic_card hcard hcop, Nat.totient_prime_pow_succ Nat.prime_two α] have hord : orderOf (ZMod.unitOfCoprime (q ^ 1) (hcop.pow_left 1)) = 2 ^ (α - 1) := by rw [← orderOf_units, ZMod.coe_unitOfCoprime, pow_one] exact orderOf_q_eq q hq5 hα rw [hord, show (2 : ℕ) - 1 = 1 from rfl, mul_one, Nat.pow_div (Nat.sub_le α 1) (by norm_num), show α - (α - 1) = 1 from by omega, pow_one]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/Factorization.lean:94-106
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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.