[[Field theory MOC]]
# Frobenius endomorphism

Let $K$ be a [[field of prime characteristic]] $p$.
The **Frobenius endomorphism** $\sigma : K \to K$ is defined by #m/def/field 
$$
\begin{align*}
\sigma(x) = x^p
\end{align*}
$$
which is indeed an endomorphism by the [[Freshman's dream]].
This is surjective, and thus a **Frobenius automorphism**, precisely when $K$ is a [[Perfect field]].

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