Show that $\mathbb F_q/\mathbb F_p$ is Galoisienne where $q=p^n$.

Question

Show that $\mathbb F_q/\mathbb F_p$ is Galoisienne where $p$ is prime and $q=p^n$ and find his Galois group. I recall that $\mathbb F_p=\mathbb Z/p\mathbb Z$.

I can't find any separable polynomial on $\mathbb F_p$ that split over $\mathbb F_q$. Any help would be helpful.

Answer 1

$X^q-X$ would do it, since its formal derivatives is $-1$ in $\mathbb{F}_p[X]$.