Sylow's theorem for Galois extensions

This fact is an application of the following pivotal fact/result/idea: Sylow's theorem
View other applications of Sylow's theorem OR Read a survey article on applying Sylow's theorem

This fact is an application of the following pivotal fact/result/idea: fundamental theorem of Galois theory
View other applications of fundamental theorem of Galois theory OR Read a survey article on applying fundamental theorem of Galois theory

Statement

Suppose $L$ is a Galois extension of a field $K$, and $G$ is the Galois group. Suppose $G$ is a finite group. Let $p$ be a prime. Then, the following are true:

1. Existence: There exists a subfield $M$ of $L$ containing $K$, such that the dimension of $M$ over $K$ is relatively prime to $p$, and the dimension of $L$ over $M$ is a power of $p$.
2. Conjugacy: Any two such subfields are conjugate inside $L$: there is a field automorphism of $L$ taking one to the other.
3. Domination: Any field such that the dimension of $L$ over it is a power of $p$, contains one such subfield $M$

Proof

The result follows directly by combining Sylow's theorem with the fundamental theorem of Galois theory.