Theorems · Theorem · number theory

IsCyclotomicExtension.Rat.zeta_sub_one_dvd_intCast_iff

∀ (p k : ℕ) [hp : Fact (Nat.Prime p)] {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
  [hK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] {ζ : K} (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))) {n : ℤ},
  hζ.toInteger - 1 ∣ ↑n ↔ ↑p ∣ n

An integer n is divisible by ζ - 1 in 𝓞 K if and only if it is divisible by p, where ζ is a primitive p ^ (k + 1)-th root of unity.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal

Cited by

1 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FactFieldNumberFieldIsCyclotomicExtension