Q) Let $E$ be a normal extension of $F$ with $[E:F] = 80$. Prove that $\exists$ a subextension $G\subset E$ s.t. $[G:F] = 5$.
Since $E$ is a normal extension, it is a splitting field of $F$. My first question is if $|Gal(E/F)| = [E:F]$ true for all $F$ or for $F$ of characteristic = $0$ like $\mathbb{Q}$? Assuming $|Gal(E/F)| = [E:F] = 80$, if $Gal(E/F)$ is a cyclic group, I know that it has a unique subgroup of order $16$ as $16|80$. Thus by Galois correspondence, $\exists$ a subextension $G$ of $E$ s.t. $[G:F] = 5$. So the second question is about proving the group is cyclic or getting the result otherwise? Thanks.