TITLE: Find separable irreducible $g$ such that $f(x)=g(x^{p^d})$
QUESTION [5 upvotes]: This is an exercise from VII.4. in Algebra: Chapter 0.

Let $\mathcal{k}$ be a field of characteristic $p$, and $f(x)\in\mathcal{k}[x]$ an inseparable irreducible polynomial. Find a separable irreducible $g\in\mathcal{k}[x]$ such that \begin{equation}f(x)=g(x^{p^d})\end{equation}for some $d\in\mathbb{N}$.

I have been looking at this problem for a while but I guess I am missing some tricks. I guess Frobenius map would come up somewhere. Also it might help to know that $f$ must be of the form \begin{equation}
f(x)=a_0+a_1x^p+a_2x^{2p}+\cdots+a_sx^{sp}.
\end{equation}  
Can someone give a hint? Thanks!

REPLY [4 votes]: Hint: Since $f$ is not separable, you can write $f(x)=g(x^p)$. If $g$ is not separable, then...