Suppose that $L$ is a field of characteristic $p$, $E$ is a field extension of $L$, a is a pth root of an element of $L$ such that a is not in $E$. Consider the polynomial $p(x):=x^p-a^p.$
Question: Let $g(x)$ be a polynomial in $E[x]$, suppose that for some $n$, $p(x)$ divides $g(x)^n$, does it follow that $p(x)$ divides $g(x)?$