Showing $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$ I want to show that $G(\mathbb{Q}(\sqrt[3]{2}, w)/ \mathbb{Q}) \simeq S_3$, but I'm not that familiar with computing Galois groups so I don't really know how to do this exercise. 
How do I approach this?
Also note that we define $w$ as: $w = \frac{1}{2}(1 + i\sqrt{3})$ 

Affirmation: If $p$ is a prime number and $f(x) \in \mathbb Q[x]$ is irreducible over $\mathbb Q$ of degree $p$ and $f(x)$ has exactly $2$ non-real roots then $\mathrm {Aut}_{\mathbb Q} L \simeq S_p$, where $L = \mathrm {Gal}  (f, \mathbb Q)$. 
With this in mind all you need to do is showing that $f(x) = x^3 -2$ is irreducible over $\mathbb Q$ and that it has $2$ complex (non-real) roots. 
