Theorems · Theorem · number theory

IsPrimitiveRoot.norm_toInteger_sub_one_eq_one

∀ {K : Type u} [inst : Field K] {ζ : K} [inst_1 : CharZero K] {n : ℕ} [IsCyclotomicExtension {n} ℚ K]
  (hζ : IsPrimitiveRoot ζ n) (h₁ : 2 < n),
  (∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n) →
    have this := ⋯;
    (Algebra.norm ℤ) (hζ.toInteger - 1) = 1

The norm, relative to ℤ, of ζ - 1 in an n-th cyclotomic extension of ℚ where n is not a power of a prime number is 1.

Defined in

Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Cited by

1 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroIsCyclotomicExtension