$X^p - t$ irreducible in $K[X]$ with $K:=\text{Quot}(\mathbb{F}_p[t])$

Question

I've heard a teacher say that this also follows by the Eisenstein criterion by checking with $t$. But then $t$ would have to be prime in order for Eisenstein to have any validity, but $t$ is arbitrary. So Eisenstein can't be used here, right?