Let $K$ be a Galois extension of $F$ and $p$ be a prime factor of the degree $[K:F]$.
I want to show that there is an intermediate extension $F\subseteq L\subseteq K$ with $[K:L]=p$.
Do we us for that the fact that the intermediate fields are in one to one correspondence with the subgroups of the Galois group?
Then since $p$ is a prime factor of the degree $[K:F]$ there must be an intermediate field with that degree according to Lagrange theorem.
Is that correct?