Let $K$ be a finite field, $f \in K[X]$ irreducible with degree $m$.
Show that $Gal(f)$ is cyclic of order $m$.
I have shown that $f$ is separable over $K$ by using that $K$ is finite and thus perfect. Thus $Gal(f)=Gal(K(a_1,...,a_m)/K)$, where the $a_i$ are pairwise distinct roots of $f$.
Furthermore since $K(a_1,...,a_m)/K$ is finite and separable, there exists a primitive element $a \in K(a_1,...,a_m)$, such that $K(a_1,...,a_m)=K(a)$.
By that $Gal(f)=Gal(K(a_1,...,a_m)/K)=Gal(K(a)/K)$.
Also since $K$ is finite, $K(a)$ is finite and thus $K(a)^\times$ is cyclic, but I am not sure if this helps.
I am stuck here. Am I on the right track? I welcome any hints.