Let $\mathbb{K}$ the the splitting field of $x^4-2x^2-2$ over $\mathbb{Q}$. Determine all the subgroups of the Galois group and give their corresponding fixed subfields of $\mathbb{K}$ containing $\mathbb{Q}$.
$\mathbb{K}$= $\mathbb{Q}(\alpha,i\sqrt{2})$ where $\alpha$ is a root. The others roots are $-\alpha, \frac{i\sqrt{2}} {\alpha},- \frac{i\sqrt{2}} {\alpha}$.
I proved that $Gal(\mathbb{K}/\mathbb{Q})$ is (isomorphic to) $D_4$ and the generators are $\sigma,\tau$ with $\sigma^4=\tau^2=id$ and $\sigma^3\tau=\tau\sigma$, for $\sigma(\alpha)=\frac{i\sqrt{2}}{\alpha}$, $\sigma(i\sqrt{2})=-i\sqrt{2}$ and $\tau(\alpha)=\alpha,\tau(i\sqrt{2})=-i\sqrt{2}$.
I considered all the subgroups of $Gal(\mathbb{Q}(\alpha,i\sqrt{2})$/$\mathbb{Q}$) but I didn't know which are the corresponding fixed subfields. What would the fixed field of {$id,\sigma^2,\sigma\tau,\sigma^3\tau$} be?