Let $F$ be a finite field of characteristic $p$ of size $p^n$ for $n \ge 1$ with the base field $K \cong Z_p$. I'm attempting to prove that the map $\phi: F → F$ sending $u$ to $u^p$ for each $u \in F$ is a $K$-automorphism of $F$ of degree $n$.

The thing is, I'm fairly certain $\phi$ is the Frobenius automorphism (correct me if I'm wrong), but doesn't that imply it's a $K$-automorphism? And if not, then I have no idea how to approach proving that it is.

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    $\begingroup$ I guess the point of the question is to show that the Frobenius automorphism is indeed an automorphism $\endgroup$ – Mathmo123 Apr 11 '16 at 20:27

You are correct that $\phi$ is the Frobenius automorphism. Of course, you have to show that it actually is an automorphism. If $\mathbb{F}$ is a field of characteristic $p$, the map $\phi : \mathbb{F} \to \mathbb{F}$ is always a field homomorphism, however it is not always an automorphism.

Show that $\phi$ is a field homomorphism. Any field homomorphism is injective. Since $F$ is finite and $\phi$ is injective, $\phi$ must then be surjective and hence $\phi$ is an automorphism.


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