Galois group of $\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$ over $\mathbb{Q}$

Question

How to find the Galois group $Gal(E|Q)$, where $E=\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})$? I know that $\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega)|\mathbb{Q})=D_3$, where $D_3$ is the dihedral group, $\omega^3=1$. Does it implies that $$\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})|\mathbb{Q})=D_3\times \mathbb{Z}_2 \quad \mbox{?}$$ Thanks for your attention!

Answer 1

Yes in this case, but no in general. What you know is $\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega,\sqrt{-1})|\mathbb{Q})$ has $D_3$ as a subgroup of index two. If $G$ is a group which has a subgroup $H$ of index two, it is not in general the case that $G$ is isomorphic to $H\times \mathbb{Z}_2$.

However, it is the case that the only group of order 12 having $D_3$ as a subgroup of index two is $D_3\times\mathbb{Z}_2$. Thus the Galois group in this case must be isomorphic to $D_3\times\mathbb{Z}_2$ (or equivalently $D_6$).