I am struggling to solve these two problems in Galois Theory. Could you help me please?
Suppose $K=\mathbb{Q}(\sqrt[5]{3}, \sqrt[5]{7})$
Prove that $[K : \mathbb{Q}]=25$
$K$ is a splitting field of the polynomial $(x^5-3)(x^5-7)$
Must consider $K$ as an extension of $\mathbb{Q}$
Find all subfields $L \subset K$ such that $[L : \mathbb{Q}]=5$
Due to Galois correspondence, we can consider subgroups of $G=Gal(K/\mathbb{Q})$ of order $5$
What are the elements in $G$ of order $5$?
I need to first analyse the structure of $G$ and find the automorphisms which generate the group, but I am not sure how to do this.