Prove that if $[L:K] = 4$ and $ \mbox{Aut}(L/K) \cong C_2 \times C_2 $ then $ L$ is of the form $K(\sqrt{a},\sqrt{b}) $.
I know that the extension is Galois, and so I can use the Galois correspondence. $C_2 \times C_2$ has 3 copies of $C_2$ as its non-trivial subgroups. These must correspond to 3 subextensions of $L/K$ of degree 2. I know also that an extension $F/K$ of degree 2 is of the form $K(c) $ for some $c^2$ in $K$. So it seems I have $ K \subseteq K(\sqrt{a}) $, $K(\sqrt{b}), K(\sqrt{c}) \subseteq L$.
From here I'm unsure what to do. It seems strange that I have three of these subextensions... I think I could do it if I had just two of them.
Thanks