Determine the subgroup of the Galois group of $\mathbb Q[i\sqrt 3]$ and $E$ the splitting field of $(t^3-2)(t^3-5)$

Question

Determine the subgroup $\text{Gal}(E/\mathbb Q[i\sqrt 3])$ where $E$ the splitting field of $(t^3-2)(t^3-5)$ and the intermediate fields. We clearly have that $E=\mathbb Q(\sqrt[3]2,\sqrt[3]5,e^{\frac{2i\pi}{3}})$, but how can I simplify this ? For the moment, I just want to get the subgroups, but I have the impression that $\text{Gal}(E/\mathbb Q[i\sqrt 3])$ is a group with $18$ elements, and so it looks very long to compute the group and the subgroups, but there is probably a trick.

Answer 1

Let's write $a=\root3\of2$, $b=\root3\of5$, $c=e^{2\pi i/3}$, and let $F={\bf Q}(c)$. Can you prove that the Galois group of $F(a)/F$ is cyclic of order 3? and the same for the group of $F(b)/F$? and then that the group of $F(a,b)/F$ is the product of these two groups?