Simple extension finder for $x^p-2$

Question

I believe that over $\mathbb{Q}[x]$, the splitting field of the polynomial $x^p-2$ is finite and separable, and so it must also be simple. However, I am unable to find, for a general $p$, what the element of the simple extension must be.

Answer 1

The splitting field is $L=\Bbb Q(\zeta_p,\sqrt[p]{2})$. A primitive element for the extension $L\mid \Bbb Q$ is $\alpha=\zeta_p+\sqrt[p]{2}$. We have $[\Bbb Q(\alpha):\Bbb Q]=p(p-1)$, which is also the degree of $[L:\Bbb Q]$.