From Wikipedia on Finite Fields:
"The polynomial $X^q-X$ factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q. This implies that, if $q = p^n$ that $X^q − X$ is the product of all monic irreducible polynomials over $GF(p)$, whose degree divides $n$. In fact, if $P$ is an irreducible factor over $GF(p)$ of $X^q − X$, its degree divides $n$, as its splitting field is contained in $GF(p^n)$. Conversely, if $P$ is an irreducible monic polynomial over $GF(p)$ of degree $d$ dividing $n$, it defines a field extension of degree $d$, which is contained in $GF(p^n)$, and all roots of $P$ belong to $GF(p^n)$, and are roots of $X^q − X$; thus $P$ divides $X^q − X$. As $X^q − X$ does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it. "
I'm confused about the significance of the fact that $X^q-X$ does not have multiple roots.