 Solution

The splitting field is the smallest subfield of $\mathbb{C}$ containing $\mathbb{Q}$ and all the roots of a polynomial under consideration. Equivalently it is generated over $\mathbb{Q}$ by those roots, and the polynomial splits there into linear factors.

Put ${\zeta }_6=e^{\pi i/3}$. All sixth roots of unity are powers of ${\zeta }_6$, so the required splitting field isIndeed ${\zeta }_6=(1+\sqrt{-3})/2$, and conversely $\sqrt{-3}=2{\zeta }_6-1$. Its field extension degree is two.