What sufficient criteria are there for a polynomial to be separable?

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.