map_rootsOfUnity — Mathlib

English

The map of roots of unity commutes with an injective monoid hom f and preserves the zpowers structure.

Русский

Образ корней единицы через инъективное моноид-мное сохраняет zpowers структуру.

LaTeX

$$map_rootsOfUnity(ζ, n) = rootsOfUnity(n, S)$$

Lean4

theorem map_rootsOfUnity {S F} [CommRing S] [IsDomain S] [FunLike F R S] [MonoidHomClass F R S] {ζ : R} {n : ℕ}
    [NeZero n] (hζ : IsPrimitiveRoot ζ n) {f : F} (hf : Function.Injective f) :
    (rootsOfUnity n R).map (Units.map f) = rootsOfUnity n S :=
  by
  letI : CommMonoid Sˣ := inferInstance
  replace hζ := hζ.isUnit_unit NeZero.out
  rw [← hζ.zpowers_eq, ← (hζ.map_of_injective (Units.map_injective (f := (f : R →* S)) hf)).zpowers_eq,
    MonoidHom.map_zpowers]

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