Let $F$ be a finite field. Let $E/F$ be a field extension of degree $m\in\mathbb{N}$. Show that $E/F$ is Galois.
I would like to varify my answer because I am absolutely sure about it;
$F$ is finite so $p=\text{char}(F)>0$. Thus $\mathbb{F}_p\subseteq F$, and this extension is finite because $F$ is finite. Suppose $n=[F:\mathbb{F}_p]$. Hence $F\cong\mathbb{F}_{p^n}$.
$$[E:\mathbb{F}_p]=[E:F]\cdot[F:\mathbb{F}_p]=m\cdot n$$ Thus, $E\cong\mathbb{F}_{p^{mn}}$ and $E$ is the splitting field of $x^{p^{nm}}-x$ over $\mathbb{F}_p$. So $E/\mathbb{F}_p$ is Galois.
Hence, $E/F$ is Galois.
Thanks in advance.