Exercise 5.1.4

Proof. The set of roots of $x^6-1$ in $ℂ$ is $S=\{1,\zeta ,{\zeta }^2,{\zeta }^3,{\zeta }^4,{\zeta }^5\}$, where $\zeta =e^{2\mathit{i\pi }∕6}=e^{\mathit{i\pi }∕3}=-{\omega }^2$. As ${\omega }^3=1$,

the splitting field of $x^6-1$ over $\mathbb{Q}$ (included in $ℂ$) is so $\mathbb{Q}(1,-{\omega }^2,\omega ,-1,{\omega }^2,-\omega )=\mathbb{Q}(S)$.

As $S\subset \mathbb{Q}(\omega )$, $\mathbb{Q}(S)\subset \mathbb{Q}(\omega )$. Conversely, $\omega \in S$, thus $\mathbb{Q}(\omega )\subset \mathbb{Q}(S)$.

Conclusion : the splitting field of $x^6-1$ over $\mathbb{Q}$ is $\mathbb{Q}(\omega )$. □