# Minimal polynomial over a field extension

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 $$(X-\alpha)$$.

My attempt:

$$X^p-\alpha^p\in F(\alpha^p)[X]$$ 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 $$E$$.