Normale extension, why $E/\mathbb F_p(t)$ is normale?
I have a theorem that says that $$E/K$$ is a normale extension if $$E$$ is the splitting field of one separable polynomial $$f(x)\in F[X]$$.
In an exercise, I have that the splitting field $$E$$ of $$X^p-t$$ is normale since $$X^p-t$$ split over $$E$$, the extension $$E/\mathbb F_p(t)$$ is normale. But $$X^p-t$$ is not separable, so the theorem doesn't hold... Any explication ?
Moreover, does $$E=\mathbb F_p(t)(\sqrt[p]t)$$ ?