How to prove that $x^p-x-t=0$ does not have a radical solution in the field $F_p(t)$?
What might be useful is that $x^p-x-t=(x-a)(x-(a+1))...(x-(a+p-1))$ in its splitting field. In addition, I have proved that if there is a radical solution, we only need to do the radical extension once. (Which can be proved using the bijection between the set of intermediate fields and the subgroups of the Galois group)
This extension is separable (compute the derivative of the polynomial). On the other hand, writing $K$ for your field, an extension of the form $K(\sqrt[p]\alpha)/K$ cannot be separable (write down the minimal polynomial and compute the derivative).