Unique subfield $L$ of $K=\mathbb{Q}(\sqrt[5]{2},\zeta_5)$ such that $[K:L]=5$

Question

$K$ is the splitting field of the polynomial $f=x^5-2$, and we need to prove the existence of a unique subfield $L$ such that $[K:L]=5$. It would be nice to use the Galois group here, but it is a semi-direct product which I am unfamilar with. Is there a way to do this without using the Fundamental Theorem of Galois Theory? I know that $[K:\mathbb{Q}]=20$, so the Galois group is a group of order $20$, but I'd like to find a way to do this without invoking the fundamental theorem. Any suggestions?

Answer 1

Following the advice of Lubin: one can show using Sylow theory that there is a unique subgroup $P$ of $G=\text{Gal}(K/\mathbb{Q})$ with $|P|=5$. Let $L=\text{Fix}(P)$. Then by Galois Theory, we have $[K:L]=|P|=5$. The uniqueness of $L$ follows from the uniqueness of $P$. Finally, $\mathbb{Q}(\zeta_5)$ is one such extension, so $L=\mathbb{Q}(\zeta_5)$.