Splitting field of x cubed minus two
= Splitting field of x cubed minus two
{title2=$\mathbb Q(\sqrt[3]{2},\zeta_3)$}
If $\alpha=\sqrt[3]{2}$ is the positive real root and $\zeta_3=e^{2\pi i/3}$, then the roots of $X^3-2$ are
$$
\alpha,\qquad \zeta_3\alpha,\qquad \zeta_3^2\alpha.
$$
Their splitting field over $\mathbb Q$ is $\mathbb Q(\alpha,\zeta_3)$.