Show that $x^{p^n}-x$ is the product of all monic irreducible polynomials in $\mathbb{Z}/p\mathbb{Z}[x]$ of a degree $d$ dividing $n$.

Question

I know that if $F$ is a field of $p^n$ elements contained in an algebraic closure $\overline{\mathbb{Z}/p\mathbb{Z}}$ of $\mathbb{Z}/\mathbb{Z}p$, then the elements of $F$ are the zeros $x^{p^n}-x$, and also that the degree of $F$ over $\mathbb{Z}/p\mathbb{Z}$ is $n$, but I don't know where to go from here. Since the elements of $F$ are the zeros of $x^{p^n}-x$ in $\mathbb{Z}/p\mathbb{Z}[x]$, that means $x^{p^n}-x$ is the product of polynomials whose zeros are the elements of $F$, but how do I know that the degree divides $n$?