 Irreducible factors of a finite-field Frobenius polynomial

Every monic irreducible polynomial over ${\mathbb{F}}_q$ whose degree divides $n$ divides $X^{q^n}-X$. In particular, if $f$ has degree $n$ and $\alpha$ is one of its roots, then ${\mathbb{F}}_q(\alpha )$ has $q^n$ elements, so ${\alpha }^{q^n}=\alpha$ and the minimal polynomial $f$ divides $X^{q^n}-X$.