Theorems · Theorem · number theory
IsPrimitiveRoot.prime_dvd_of_dvd_norm_sub_one
∀ {n : ℕ} (hn : 2 ≤ n) {K : Type u_1} [inst : Field K] [NumberField K] {ζ : K} {p : ℕ} [hF : Fact (Nat.Prime p)]
(hζ : IsPrimitiveRoot ζ n), ↑p ∣ (Algebra.norm ℤ) (hζ.toInteger - 1) → p ∣ nLet ζ be a primitive root of unity of order n with 2 ≤ n. Any prime number that divides the
norm, relative to ℤ, of ζ - 1 divides also n.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 220 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldNumberFieldFact
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Cites53
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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Cited by1
Results whose statement or proof uses this declaration.
- IsPrimitiveRoot.not_coprime_norm_of_mk_eq_oneproof · cited by 1