Find the subfield corresponding to certain subgroup of Galois group

Question

Let $\alpha = \sqrt{2 + \sqrt{5}}, \beta = \imath\sqrt{\sqrt{5} - 2}$. The Galois group of $\mathbb{Q}(\alpha, \beta):\mathbb{Q}$ is isomorphic to $D_4$. Find subfield corresponding to subgroup (cyclic subgroup of order $4$) given by permutations $\{id, (1324), (12)(34), (1423)\}$, where $1$ represents $\alpha, 2$ represents $-\alpha, 3$ repr. $\beta, 4$ repr. $-\beta$. I was able to find subfields corresponding for all the other subgroups, but I'm stuck here.

Answer 1

The automorphism of order $4$ maps $\alpha\to\beta\to-\alpha$. It maps $\sqrt5=\alpha^2-2$ to $\beta^2-2=-\sqrt5$ and $i=\alpha\beta$ to $-\alpha\beta=-i$. It fixes $\sqrt{-5}$. The fixed field is $\Bbb Q(\sqrt{-5})$.