Exercise 6.4.14

Proof. Let $L=\mathbb{Q}({\zeta }_p,\sqrt[p]{2})$.

$\sqrt[p]{2}\in L$, and ${\zeta }_p\sqrt[p]{2}\in L$, thus $\mathbb{Q}(\sqrt[p]{2},{\zeta }_p\sqrt[p]{2})\subset L$.

${\zeta }_p={\zeta }_p\sqrt[p]{2}∕\sqrt[p]{2}\in \mathbb{Q}(\sqrt[p]{2},{\zeta }_p\sqrt[p]{2})$, and $\sqrt[p]{2}\in \mathbb{Q}(\sqrt[p]{2},{\zeta }_p\sqrt[p]{2})$. As $L$ is the smallest subfield of $ℂ$ containing $\mathbb{Q}$, ${\zeta }_p,\sqrt[p]{2}$, then $L\subset \mathbb{Q}(\sqrt[p]{2},{\zeta }_p\sqrt[p]{2})$.

Conclusion :

The splitting field of $x^p-2$ over $\mathbb{Q}$ is generated by two of its roots. □