Consider the cubic polynomial $$ f = x^3+px+q, $$ where $p,q$ are elements of a fixed algebraic closure $\overline{\mathbb{F}}_2$ of $\mathbb{F}_2$.

Is there an elegant criterion for deciding whether $f$ has $0$, $1$ or $3$ roots in the field $\mathbb{F}_2(p,q)$?

For example, I would consider as "elegant" any criterion stipulating that certain polynomial expressions in $p,q$ be "of a certain form," e.g., are themselves values of certain polynomials (with coefficients in $\mathbb{F}_2$).