What is the splitting field of $$(X^3-2)(X^3-3)(X^2-2)\ \ ?$$
The roots are $$\{\sqrt[3]2,\ \ j\sqrt[3]2,\ \ j^2\sqrt[3]2,\ \ \sqrt[3]3,\ \ j\sqrt[3]3,\ \ j^2\sqrt[3]3,\ \ \sqrt 2,\ \ -\sqrt 2\}$$ where $j=e^{\frac{2i\pi}{3}}$. So it splitting field is included in $$\mathbb Q(j,\sqrt[3]2,\sqrt[3]3,\sqrt 2)$$ but is all the field ?
Yes. It is clear that $(X^3-2)(X^3-3)(X^2-2)$ splits over $\mathbb Q(j,\sqrt[3]2,\sqrt[3]3,\sqrt 2)$. On the other hand, any splitting field of $(X^3-2)(X^3-3)(X^2-2)$ over $\Bbb{Q}$ must contain $\sqrt[3]2,\sqrt[3]3,\sqrt 2$ and $\frac{j\sqrt[3]2}{\sqrt[3]2}=j$.