Show that the polynomial $x^{q^n}-x$ does not split over $\Bbb F_q$

Question

Show that the polynomial $x^{q^n}-x$ does not split over $\Bbb F_q$.

I don't know where should I start. Should I rewrite and expand $x^{q^n}-x$?

Answer 1

Recall that a polynomial $f(x)$ over a field $k$ has multiple roots (over the algebraic closure of $k$) if and only if it is not coprime with its derivative. Since this polynomial's derivative is identically $-1$, it is automatically coprime with its derivative, so each of its $q^n$ roots over the algebraic closure are distinct. But the field $\mathbb{F}_q$ only contains $q$ elements, so this polynomial cannot split over $\mathbb{F}_q$ so long as $n > 1$.