Here is a lemma I encountered here: https://imathworks.com/math/math-why-xpn-x1-is-irreducible-in-mathbbf_p-only-when-n1-or-np2/
Lemma: If $f(x)$ is irreducible in $F_p[x]$, Then for any positive integer k: $$f(x)\mid x^{p^k}-x \Leftrightarrow \deg(f) \mid k $$
I think I can show "$\Rightarrow $" part as follows:
Suppose $f(x)\mid x^{p^k}-x$. Let $L$ be the splitting field of $x^{p^k}-x$ over $F_p$, $\alpha$ be a root of $f(x)$, Then: $$F\subset F(\alpha)\subset L $$ By tower theorem: $$k=[L:F]=[L:F(\alpha)]\cdot [F(\alpha):F]=\deg(f)\cdot m $$
But I cannot show the other side. Is there any hints? Thanks.