 Solution

Write $g(x)=x^6+x^3+1$. The second polynomial is $g(-x)$. Negating the roots of a polynomial does not change the field generated by them: each original root is minus a new root, and conversely. Hence this splitting field is again $\mathbb{Q}({\zeta }_9)$ and its field extension degree is six. Equivalently, it is the cyclotomic field $\mathbb{Q}({\zeta }_{18})=\mathbb{Q}({\zeta }_9)$, since ${\zeta }_{18}=-{\zeta }_9^5$.