= Solution
The <polynomial identity>
$$
x^9-1=(x^3-1)(x^6+x^3+1)
$$
shows that the roots are the ninth <roots of unity> which are not third roots. These are precisely the primitive ninth <roots of unity>, all powers of $\zeta_9=e^{2\pi i/9}$. Thus the <splitting field> is $\mathbb Q(\zeta_9)$. The supplied <irreducible polynomial>[irreducibility] statement 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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