English
If L/K is an algebraic extension, the Frobenius K-algebra endomorphism of L is an automorphism of L over K.
Русский
Если расширение L/K является алгебраическим, то Фробениус-алгебра-одномерность на L над K является автоморфизмом L над K.
LaTeX
$$frobeniusAlgEquivOfAlgebraic K L is an AlgEquiv K L L$$
Lean4
/-- If `R` is a perfect ring and an algebra over a finite field `K`, the Frobenius `K`-algebra
endomorphism of `R` is an automorphism. -/
@[simps!]
noncomputable def frobeniusAlgEquiv (p : ℕ) [ExpChar R p] [PerfectRing R p] : R ≃ₐ[K] R :=
.ofBijective (frobeniusAlgHom K R) <| by
obtain ⟨p', _, n, hp, card_eq⟩ := card' K
rw [coe_frobeniusAlgHom, card_eq]
have : ExpChar K p' := ExpChar.prime hp
nontriviality R
have := ExpChar.eq ‹_› (expChar_of_injective_algebraMap (algebraMap K R).injective p')
subst this
apply bijective_iterateFrobenius
