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

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

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 $\mathbb{F}_2$).