Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension.
1 How do I prove that $EL$ is a finite field extension of $E$?
2 How do I prove that the natural restriction map Gal($EL/E)\rightarrow$ Gal($L/(L\cap E)$) is an isomorphism?
So I was thinking about using Galois main theorem for this, but Galois is very new to me and I'm not sure how to look at this. I would greatly appreciate any nudges in the right direction.
For the first part you don't need that $L/A$ is Galois : take a basis of $L/A$ and try to show that this gives you a generating set for $EL/E$.
Then try to see why $EL$ is at least a domain, and why an integral finite-dimensional algebra over a field must be a field.