Sylow's theorem for Galois extensions

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.