Irreducible polynomial over a field $k$ with $char\ k = p > 0$

Question

I'm studying for my Abstract Algebra II final and reviewing problems. I'm having some trouble with this one. Direction would be helpful.

Let $k$ be a field with $char\ k = p > 0$, and let $f(x) \in k[x]$ be an irreducible polynomial. Prove that there exists an integer $n \geq 0$ and an irreducible separable polynomial $g(x) \in k[x]$ such that $f(x) = g(x^{p^n})$.

EDIT: I have found a proof here, but do not understand it fully. It is Lemma 4.3. Clarification? https://math.berkeley.edu/~amathew/chfields.pdf

Answer 1

All you have to know is a polynomial $f$ is separable if and only if $(f,f')=1$. If $f$ is irreducible and $f'\neq 0$, it is always the case that $(f,f')=1$. In characteristic $0$, $f$ is always separable.

Thus $f$ is not separable if and only if the characteristic is $p>0$ and $f$ is a polynomial in $x^p$.

Let $n$ be the greatest exponent such that $f(x)$ is a polynomial in $x^{p^n}$: $\,f(x)=g\bigl(x^{p^n}\bigr)$. $g$ is irreducible since $f$ is, and it is not a polynomial in $x^p$ by the maximality of $n$, hence it is separable.