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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 $E=\mathbb F_p(t)(\sqrt[p]t)$ ?