Let K be a finite field and $f\in K[X] $ and irreducible with $deg(g)=m$. Show (1) f is seperable over K and (2) $Gal(f)$ is cyclic with $|Gal(f)|=m$.
In the following L denotes the splitting field of f over K.
(1) K is finite $\implies$ K is perfect $\implies$ f is seperable over K
(2) K is perfect and $L/K$ is a finite field extension $\implies \exists a\in L$ primitive element so that $L = K(a)$. Ok then $|Gal(f)|=m$.
But how to prove that Gal(f) is cyclic?
I know that every zero of f is in the form: $k_1 + k_2a^i$ with $ i\in \{1,...,m-1\}$ and $k_1,k_2\in K$. That's where i am stuck...