= 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> is
$$
\boxed{\mathbb Q(\zeta_6)=\mathbb Q(\sqrt{-3})}.
$$
Indeed $\zeta_6=(1+\sqrt{-3})/2$, and conversely $\sqrt{-3}=2\zeta_6-1$. Its <field extension degree> is two.
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