Let $L := \Bbb Q(\sqrt 5,\sqrt 7).$ We have proved that $L/\Bbb Q$ is galois.
I have to find all the intermediate fields of $L/\Bbb Q$.
So far, I have found $\Bbb Q(\sqrt 5)$, $\Bbb Q(\sqrt 7)$, $\Bbb Q(\sqrt{35})$ and $\Bbb Q\left(\sqrt{\frac57}\right)$.
To prove, that these are all intermediate fields:
Let $L \supset E\supset \Bbb Q$ be a intermediate field. Then $[E:\Bbb Q]$ has to divide $[L:\Bbb Q]=4$. If $[E:\Bbb Q]=1$, then $E=\Bbb Q$. If $[E:\Bbb Q]=4$, then $E=L$. So $[E:\Bbb Q]$ has to be $2$.
So my questions are:
- Are these all intermediate fields? If no, which have I missed?
- How do I prove that $E$ has to be one of the above fields? Any hints are welcome.
Best Answer
What is the Galois group of the extension? What does the Galois correspondence say about intermediate fields?