$K \subset M \subset L \Rightarrow$ Gal$(L/M) < $ Gal$(L/K)$

Question

Let $K \subset M \subset L$ be field extensions.

How to show that Gal$(L/M) < $ Gal$(L/K)$?

I tried:

Since $M$ is the fixed field of $L$ under $H$ and $K$ is fixed by $H$, so $K$ is a subfield of $M$.

I don't know if it's correct and how to continue.

How to show it right?

Answer 1

It follows from the definitions. If $\sigma\in Gal(L/M)$ then $\sigma$ is an automorphism on $L$ which fixes every element of $M$. But since every element of $K$ is also an element of $M$ it follows that $\sigma$ fixes all the elements of $K$. Hence $\sigma\in Gal(L/K)$.