I know that $p$ is separable when the discriminant is not zero, $p'$ and $p$ don't have a common root, $p$ is irreducible, but that's it. Is there a criterion over $\mathbb Q$ for separability in terms of the coefficients similar to Eisenstein or related criteria for irreducibility?
I am asking this since my conjecture is that polynomials $1+\sum_{a} t^a$ seem to be separable where $a$ runs through a nonempty set of distinct odd numbers.
Thanks for your help.