I came across this question while doing some exercises at the end of the chapter. I would like someone to comment on my solution (on its correctness, completeness and approach):
Let $\alpha\in E$, where $E$ is an algebraic extension of a field $F$ with characteristic $p$. Let $m(X)$ be the minimal polynomial of $\alpha$ over the field $F(\alpha^p)$. Show that $m(X)$ splits over $E$, and in fact $\alpha$ is the only root, so that $m(X)$ is a power of $(X-\alpha)$.
$X^p-\alpha^p\in F(\alpha^p)[X]$ is one of the polynomials of which $\alpha$ is a root and we know that $m(X)|X^p-\alpha^p$.
Now if we consider $X^p-\alpha^p\in E[X]$, then we can write it is $(X-\alpha)^p$. And so as $m(X)|(X-\alpha)^p$, $m(X)$ is a power of $X-\alpha$ and splits over $E$.