Computing the Galois Group and intermediate fields for the extension $\mathbb{Q}(3^{1/4}, \eta_3)/\mathbb{Q}$

Question

Suppose that $\eta_3$ is a primitive third root of unity and consider the extension of fields $\mathbb{Q}(3^{1/4}, \eta_3)/\mathbb{Q}$. I'd like to compute the Galois Group $\mbox{Aut}_\mathbb{Q} \,\mathbb{Q}(3^{1/4}, \eta_3)$ (where $3^{1/4} \in \mathbb{R}$) and its intermediate fields. I'm familiar with how to do this with splitting fields for polynomials, but I'm not sure how to approach this particular problem.

Answer 1

According to RJM's remark above the field is $\mathbb{Q}(\sqrt[4]{3},i)$ now

$$\mathbb{Q}(\sqrt[4]{3},i)/\mathbb{Q}(i)$$ is Galois with automorphism $$\tau(\sqrt[4]{3})=i\sqrt[4]{3}$$

And $$\mathbb{Q}(\sqrt[4]{3},i)/\mathbb{Q}(\sqrt[4]{3})$$ is also Galois with automorphism $$\sigma(i)=-i$$ It is easy to calculate that $\sigma$ and $\tau$ do not commute. In fact you can find the relation $$\sigma\tau\sigma=\tau^3$$. This identifies the group as the dihedral group $D_8$.