Past exam of the number theory course of the University of Oxford / 2011 / A / 1 / b / i / Solution

Factorization over the rational numbers givesThe rational roots add nothing to the splitting field. By Eisenstein's criterion at $7$, $x^3-7$ is irreducible. Put $a=\sqrt[3]{7}$ and $\zeta =e^{2\pi i/3}$. Its splitting field is $L=\mathbb{Q}(a,\zeta )$. The real field $\mathbb{Q}(a)$ has degree $3$; the nonreal element $\zeta$ has degree $2$ over it. Thus $[L:\mathbb{Q}]=6$, and its faithful action on the three nonrational roots identifies