Let $K = \mathbb Q(\alpha)$, for $\alpha$ a root of $a^4 + 4 \alpha^2 + 2 = 0$. I want to prove the group of units $\mathcal O_K^*$ equals $\langle -1, \alpha^2 + 1\rangle$.
I've found the ring of integers $\mathcal O_K$ is $\mathbb Z[\alpha]$, and that the number field has a trivial class group. Dirichlet's Unit Theorem tells me that since the number of pairs of complex embeddings is 2, I have that the group of units is of rank 1, i.e. $\mathcal O_K^* = \langle -1, \varepsilon \rangle$.
I suspect that the torsion group $\mu(K)$ equals $\langle -1 \rangle$ because we can embed $\alpha \mapsto \sqrt{-2 + \sqrt{2}} \in \mathbb C$, which has absolute value greater than 1, but I'm not sure whether that reasoning is correct.
One result that might be helpful is that I can prove that for any unit $\nu$, I have that $\nu^2$ is of the form $a + b \alpha^2$ for integers $a, b \in \mathbb Z$, but I tried exploring that and it didn't really lead anywhere.