Let $k$ be a field with characteristic $p \gt 0$ and let $q(x) = x^p-ax-b \in k[x]$ be a polynomial with $a \neq 0$. Let $K$ be a splitting field of $q(x)$. I have to prove that $x^{p-1}-a$ splits in $K$.
What I was able to do - If we assume $\alpha$ is a root of $x^{p-1}-a$, then $a\alpha$ is also its root. Now all I have to prove is that $\alpha \in K$. This part actually seems easy but somehow I am unable to tackle this problem.