Extension of a finite field

Question

I'd like a hint for the following problem: Let $p<q$ be prime numbers, $p$ does not dividing $q-1$. Show that there is a extension $L$/ $\mathbb{Z}_q$. Which is the splitting field extension for each of the polynomials $x^p - a$ .Where $a$ in the multiplicative group of $\mathbb{Z}_q$.thanks.

Answer 1

Let $L=\mathbb{F}_q(\zeta_p)$, i.e. the splitting field of $x^p-1 \in \mathbb{F}_q[x]$. Choose $u,v \in \mathbb{Z}$ with $up + v(q-1)=1$. For every $a \in \mathbb{F}_q^*$ we have $a=(a^{u})^p (a^{q-1})^v=b^p$ with $b=a^u$. It follows that $x^p-a=\prod_{i=0}^{p-1} (x-\zeta_p^i b)$ splits over $L$.