I want to prove the following statements. I'd like some feedback on part (a) and maybe a hint for part (b).
Let $K$ be a Galois extension of $F$ with $|\text{Gal}(K/F)|=20$.
(a) Prove that there exists a subfield $E$ of $K$ containing $F$ with $[E:F]=5$.
Proof. As $G=\mathrm{Gal}(K/F)$ is a group of order $20=2^2\cdot 5$, by Sylow's theorem, there is a Sylow 2-subgroup $H$ of $G$, i.e., $[G:H]=5$ and so, by the fundamental theorem of Galois Theory, there is an intermediate field $E$ with degree $5$ over $F$.
(b) Determine whether there must also exist a subfield $L$ of $K$ containing $F$ with $[L:F]=10$.
Best Answer
For (b)
By Cauchy's theorem there is an element of order 2 in $G$. Let $H$ be the subgroup generated by this element. Thus $[G:H]=10$, so by the fundamental theorem there exists subfield $L$ such that $[L:F]=10$.