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 \b[six]. Equivalently, it is the <cyclotomic field> $\mathbb Q(\zeta_{18})=\mathbb Q(\zeta_9)$, since $\zeta_{18}=-\zeta_9^5$.

Solution

Past exam of the options I course of the University of Oxford / 2009 / A / A1 / ii / Solution

 Solution (source code)