Originally Posted by

**Carl** Ok, I think I got it now:

So, I say that $\displaystyle Gal(L_2/ \mathbb{Q})=V$

And since V has 3 subgroups of order 2, which are $\displaystyle \langle \sigma\rangle, \langle \tau \rangle$ and $\displaystyle \langle \sigma\tau\rangle$, then there are 3 $\displaystyle \alpha$ such that $\displaystyle [\mathbb{Q}(\alpha):\mathbb{Q}]=2$ and those are $\displaystyle \sqrt{2}i, i $ and $\displaystyle \sqrt{2}$ since the splitting field for $\displaystyle x^4+1$ is $\displaystyle \mathbb{Q}(\sqrt{2},i)$.

And then I just have to check that those 3 are independent from each other and thereby I have found the 3 $\displaystyle K_2$'s

Did I get that right?