Let $x$ be one of the nontrivial cubic roots of the unity and let $y$ be the cubic root of $2$.
Prove that $\mathbb{Q}(x,y)$ is a Galois extension and then, show that is isomorphic to one of the groups of order $6$ by giving explicit isomorphisms.
I already know that the degree of the extension is $6$, so i don´t know if its isomorphic to $S_3$ or, $\mathbb{Z}_2\times \mathbb{Z}_3$, and obviously I don´t know how to make the correspondence.
Aditional, how I can prove that is a Galois extension?
Possible sketch: Show that $\mathbb{Q}(\sqrt[3]{2},\omega)$ is the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Show that the Galois group is of order six. Show that there are automorphisms $\sigma : \omega \mapsto \omega^2$ and $\tau : \sqrt[3]{2} \mapsto \sqrt[3]{2}\omega$. The first is order $2$ and the second is order $3$ and hence together must generate the group. Check if these commute, and decide correspondingly whether the group is $\mathfrak{S}_3$ or $\mathbb{Z}/2 \times \mathbb{Z}/3$.