Galois extension of two fields

Question

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it?

For fields $K\subseteq L,M\subseteq \bar K$ we denote by $LM$ the smallest subfield of $\bar K$ containing $L$ and $M$. Assume that $L$ is the Galois extension of field $K$. Show that:

1. $LM$ is the Galois extension of $M$;
2. Galois group $G(LM/M)$ is isomorphic to the subgroup Galois group $G(L/K)$;
3. If additionally $(L:K)=(LM:M)$, then groups $G(LM/M)$ and $G(L/K)$ are isomorphic.

I appreciate any kind of help.

Thanks