Originally Posted by

**roninpro** I am trying to compute the Galois group for the polynomial $\displaystyle x^4-6x^2+7$, but I'm having a bit of trouble. I know that $\displaystyle x^4-6x^2+7=(x^2-(3-\sqrt{2}))(x^2-(3+\sqrt{2}))$ over $\displaystyle \mathbb{Q}(\sqrt{2})$. The polynomial will then split over $\displaystyle \mathbb{Q}(\sqrt{2},\sqrt{3+\sqrt{2}},\sqrt{3-\sqrt{2}})$, which is an extension of degree 8.

Now I have the automorphisms $\displaystyle \tau_1: \sqrt{2}\mapsto -\sqrt{2}$, $\displaystyle \tau_2: \sqrt{3+\sqrt{2}}\mapsto -\sqrt{3+\sqrt{2}}$ and $\displaystyle \tau_3: \sqrt{3-\sqrt{2}}\mapsto -\sqrt{3-\sqrt{2}}$. I know that $\displaystyle \tau_1^2=\tau_2^2=\tau_3^2=e$. But I can't seem to get relations between them (and in particular, I would like to find an element of order 4 to show that this is the dihedral group). For example, I am interested in powers of $\displaystyle \tau_1 \tau_2$ but I'm not sure how to run the computation.

Any suggestions would be much appreciated.