Existence of an intermediate field $K\subseteq M \subseteq L$ such that $[L:M]=p$

Question

Let $L/K$ be a finite galois extension (normal and separable).

Let $p$ be a prime number which divides $[L:K]$.

Is there necessarily an intermediate field $K\subseteq M \subseteq L$ such that $[L:M]=p$?

any help would be appreciated...

Answer 1

By Cauchy's theorem, there exists an element $x$ of order $p$ in the Galois group, and the subgroup $\langle x\rangle$ generated by $x$ has order $p$. The Galois correspondence gives us the intermediate field. More generally, for each $k$ such that $p^k\mid[L:K]$ we have a subgroup of order $p^k$ (this is a generalization of a Sylow theorem which in a textbook would usually be presented at the same time), hence an intermediate field $M$ with $[L:M]=p^k$.