Irreducible factors of a finite-field Frobenius polynomial
= Irreducible factors of a finite-field Frobenius polynomial
{title2=$X^{q^n}-X$}
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 $X^{q^n}-X$.