Galois group of $x^4+2x^2-2$ and intermediate fields.

Let $$\alpha_i,\; 1 \leq i \leq 4$$ be the four roots of the polynomial. I know that the Galois group is isomorphic to $$D_4$$. To find the intermediate fields of the splitting field of the polynomial, I find the fixed fields of the corresponding subgroups. However I was not able to find the intermediate field corresponding to $$H = \langle r_{180} \rangle$$ (the rotation by $$180^\circ$$). Any hints or solutions would be appreciated.

• That subgroup is the intersection of all the index two subgroups of $D_4$. Therefore the fixed field is the compositum of the quadratic intermediate fields. Have you identified those? Commented Mar 18, 2021 at 21:40
• @JyrkiLahtonen thank you very much,I understand your reasoning, it seems I just need to research dihedral groups a bit more. Commented Mar 18, 2021 at 22:31
• I assume you have a list of zeros of the quartic. Their form yields an obvious quadratic subfield, and using their products you find another. It is difficult to give a more helpful hint because you did not share everything you know about this splitting field. Taken somewhat literally, you gave the impression that you have a description of the intermediate fields corresponding to other subgroups. Is that correct? Commented Mar 19, 2021 at 4:57
• @JyrkiLahtonen yes I have a list of all intermediate fields and subgroups, I just could not figure out, how to use galois correspondence to arrive at the fixed field for the rotation by 180 degrees. Your initial comment has already explained how to. Commented Mar 19, 2021 at 14:54

Hint: $$x^4 + 2 x^2 -2 = (x^2+1)^2 - 3.$$
• @MathR from the hint, we have the polynomial splitting over the $\sqrt3$ extension as $(x^2+1+\sqrt3)(x^2+1-\sqrt3)$. Commented Apr 18, 2021 at 18:15