Today I am studying the extension $\mathbb{Q}(\xi_{12})/\mathbb Q$, more precisely I want to study its subfield and associated subgroups in the Galois group. I already know that $ G=\textrm{Gal}(\mathbb{Q}(\xi_{12}):\mathbb Q ) \cong (\mathbb Z/12\mathbb Z)^\times \cong \mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z $ and so the nonzero proper subgroups of the Galois group $G$ are the three proper subgroups of $\mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z$.
How do I write explicitly the corresponding subfields in $\mathbb Q(\xi_{12})$ of these subgroups? Is there a fixed technique to do so?
Thanks!
Among the twelfth roots of unity are the cube roots of unity and the fourth roots of unity. A $\xi_3$ is $-\frac12+\frac{\sqrt{-3\,}}2$, and $i$ is a fourth root. I think you can spot the other quadratic extension of $\Bbb Q$ in $\Bbb Q(\sqrt{-3},i)$.