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.
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.