= Solution
{c}
The <splitting field> in $\mathbb C$ is the smallest subfield containing $\mathbb Q$ and every <root> of the <polynomial>. The sixth roots of unity are powers of $\zeta_6=e^{i\pi/3}$, so the <splitting field> of $x^6-1$ is $\mathbb Q(\zeta_6)=\mathbb Q(\sqrt{-3})$.
For $x^6+x^3+1=(x^9-1)/(x^3-1)$, the roots are exactly the primitive ninth roots of unity, all powers of $\zeta_9=e^{2\pi i/9}$. Thus its <splitting field> is $\mathbb Q(\zeta_9)$. The supplied irreducibility makes the displayed degree-six <polynomial> the <minimal polynomial> of $\zeta_9$, giving
$$
\boxed{[\mathbb Q(\zeta_9):\mathbb Q]=6.}
$$
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