iterateFrobenius_algebraMap — Mathlib

English

On the bottom field K, the iterateFrobenius map acts as x ↦ x^{p^{n}} on elements of K, under the exponent bound hn.

Русский

На нижнем поле K итеративный Фробениус действует как x ↦ x^{p^{n}} на элементах K при ограничении экспоненты hn.

LaTeX

$$$\text{iterateFrobenius}_{K,L}(p,n) : K \to K$ satisfying algebraMap_KL( iterateFrobenius a ) = a^{p^{n}}$$$

Lean4

/-- Action of `iterateFrobenius` on the bottom field. -/
theorem iterateFrobenius_algebraMap {n : ℕ} (hn : exponent K L ≤ n) (a : K) :
    iterateFrobenius K L p hn (algebraMap K L a) = a ^ p ^ n :=
  by
  apply (algebraMap K L).injective
  rw [map_pow, algebraMap_iterateFrobenius K p hn]

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