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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.

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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.

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  • $\begingroup$ How would I show that for the first part? And how should I handle the second part? $\endgroup$ – user314965 Apr 16 '16 at 13:18

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