It is a two part question.
Let $K/F$ be a Galois extension with $[K : F] = n$. If $p$ is a prime divisor of $n$, prove there is an intermediate field $L$ with $[K : L] = p$. Prove or disprove that there is an intermediate field $M$ with $[M : F] = p$.
Could anyone help me for some hints please, Thanks a lot!