Let $G$ be a finite group and $K$ a finite field. There exists $L$ finite field such that $L/K$ is Galois and $\mathrm{ Gal} (L/K) \simeq G$?

Question

My question is

Let $G$ be a finite group and $K$ a finite field. Is true that always exists another finite field $L$ such that $L/K$ is Galois and $\mathrm{ Gal}\left(L/K\right) \simeq G$?

I tried to solve it or to find a counterexample, but my problem is that I don't know what to do with the general group $G$.

Thank you in advance.

Answer 1

The Galois group of a finite extension of finite fields is always cyclic, generated by the Frobenius $\sigma : X\mapsto X^q$ if $K\cong\Bbb F_q$. So any noncylic group $G$ cannot be obtained as $\operatorname{Gal}(L/K)$ for $L/K$ a finite extension of finite fields.