Galois theory and the connection to prime numbers

Question

Let $F \subseteq E $ be a finite Galois extension with $G =\operatorname{ gal}( E : F )$. If $p$ is a prime integer and $p \mid [ E : F ] $, show that there exists an intermediate field extension $F \subseteq K \subseteq E$ such that $[E:K] = p$.

Answer 1

One has $|G|=[E:F]$, hence using Cauchy's theorem, there exists $H\leqslant G$ of cardinality $p$. Therefore, using Galois' correspondence, there exists an intermediate field $K(=E^H)$ such that $[E:K]=|H|$.