Given a quadratic equation $f(x) =x^2 + bx +c$, with rational coefficients $b,c$ a sufficient condition for $f$ to have a rational root is for $\Delta = b^2 - 4c$ to be a square in $\mathbb{Q}$.
I was curious, does there exist a similarly simple sufficient condition one can check to see if the depressed cubic $g(x) = x^3+px+q$ with $p,q \in \mathbb{Q}$ has at least one rational root?