Subgroups of a Galois Group

Question

I've just started studying Galois Theory and I'm having a litte trouble with the following exercise:

Find all the subgroups of $\operatorname{Gal}\big(X^4-X^2 -2\ ;\mathbb{Q}\big)$. Which of the subgroups are normal?

I appreciate the help. Thanks

Answer 1

You can solve for the roots of $Y^2-Y-2$ which are $2$ and $-1$, so that the roots of your polynomial are $\pm \sqrt 2$ and $\pm i$. Then you're trying to find the Galois group of $\mathbb Q(i,\sqrt 2)/\mathbb Q$, which is $C_2\times C_2$. Since this is abelian, all subgroups are normal.