The statement of the exercise from the textbook goes:
Let $a \in E$, where $E$ is an algebraic extension of a field $F $ of prime characteristic $p$. Let $m(X)$ be the minimal polynomial of $a$ over the field $F(a^p)$. Show that $m(X)$ splits over $E$, and in fact $a$ is the only root, so that $m(X)$ is a power of $(X - a)$.
I can do: $m(x)$ divides $x^p-a^p= (x-a)^p$ by the freshman’s dream and for $p$ an odd prime, so $m(x)= (x-a)^k$ for some $k\leq p$ and $a$ is the only root and $m(x)$ splits over $E$.
Is there a way to show first that $m(x)$ splits over $E$ before doing what I did? And does this apply still for $p=2$?