Finding the Galois group of $\mathbb Q (\sqrt 5 +\sqrt 7) \big/ \mathbb Q$ I know that this extension has degree $4$. Thus, the Galois group is embedded in $S_4$. I know that the groups of order $4$ are $\mathbb Z_4$ and $V_4$, but both can be embedded in $S_4$. So, since I know that one is cyclic meanwhile the other is not, I've tried to determine if the Galois group is cyclic but I couldn't make it. Is there any other way?

You should first prove that $\mathbf{Q}(\sqrt{5}+\sqrt{7})/\mathbf{Q}$ is a Galois extension. For this it may be useful to verify that $\mathbf{Q}(\sqrt{5}+\sqrt{7}) = \mathbf{Q}(\sqrt{5},\sqrt{7})$. Then you might consider the Galois groups of $\mathbf{Q}(\sqrt{5})/\mathbf{Q}$ and $\mathbf{Q}(\sqrt{7})/\mathbf{Q}$.
