norm_toInteger_sub_one_of_eq_two — Mathlib

English

If s ≤ k and p ≠ 2, then the norm of (ζ^{p^s} − 1) equals p^{p^s}.

Русский

Если s ≤ k и p ≠ 2, норма (ζ^{p^s} − 1) равна p^{p^s}.

LaTeX

$$$\\operatorname{norm}_{\\mathbb{Z}}\\bigl(h\\zeta^{p^{s}} - 1\\bigr) = p^{p^{s}}$$$

Lean4

/-- The norm, relative to `ℤ`, of `ζ - 1` in a `2`-th cyclotomic extension of `ℚ` is `-2`.
-/
theorem norm_toInteger_sub_one_of_eq_two [IsCyclotomicExtension { 2 } ℚ K] (hζ : IsPrimitiveRoot ζ 2) :
    norm ℤ (hζ.toInteger - 1) = -2 :=
  by
  rw [show 2 = (2 ^ (0 + 1)) by norm_num] at hζ
  simpa using hζ.norm_toInteger_pow_sub_one_of_two

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