Let $K$ be a field with $\operatorname{char}K =p$, suppose that $x^p- a$ is irreducíble over $K$ and let $\alpha$ a root of $x^p-a$. Prove that there is no roots of $x^p - \alpha$ in $K(\alpha)$.
$\textbf{My attempt:}$
It's easy to see that if $\lambda$ is a root of $x^p - \alpha$ then $\lambda$ is the only root of $x^p- \alpha$ (because, $\operatorname{char}K =p$). So, we just need to prove that $\lambda \notin K(\alpha)$, but I can't do that, can you help me??
Assume $\lambda^p=\alpha$, and write $$\lambda=\sum_{i=0}^{p-1}k_i\alpha^i,$$ where $k_i\in K$. What happens if you try to expand $\lambda^p$?