Theorems · Theorem · number theory

IsCyclotomicExtension.Rat.isCoprime_of_not_zeta_sub_one_dvd

∀ (p : ℕ) [hp : Fact (Nat.Prime p)] {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
  [hK : IsCyclotomicExtension {p} ℚ K] {ζ : K} (hζ : IsPrimitiveRoot ζ p) {x : NumberField.RingOfIntegers K},
  ¬hζ.toInteger - 1 ∣ x → IsCoprime (↑p) x

If ζ - 1 does not divide x, then p and x are coprime, where ζ is a primitive p-th root of unity and p is prime.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal

Cited by

0 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FactFieldNumberFieldIsCyclotomicExtension