The full question is as follows: Take $M=\mathbb{Q}(i,\sqrt{2})$ and $\alpha=1+i+\sqrt{2}$. Prove that $G=Aut(M)$ is isomorphic to $V_4$, where $V_4$ is the group of order 4 not cyclic (Klein group).
I’ve seen a few examples of this type of problem. I understand the method is to first prove that $G$ has order 4, and then that none of its elements generate the group (hence it’s not cyclic and therefore it is isomorphic to $V_4$.
What gets me in trouble is proving it has degree 4, since I am not allowed to use advanced results of Galois theory, only some theorems about field extensions.
Second part of the problem says: Prove that $f=\Pi_{\sigma\in G} (X-\sigma(\alpha))$ is the minimum polynomial of $\alpha$ over $\mathbb{Q}$.
Some ideas on how to attack this problem?