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 $\mathbb{F}_2$).
There are $1$ or $3$ roots if and only if $q$ is in the value set of $x^3+px$ on $K=\mathbb F_2(p,q)$. It remains to decide between these two cases, which can be done using the Berlekamp discriminant: The case $q=0$ is trivial, so assume that $q\ne0$, hence $f$ is separable. Let $\alpha,\beta,\gamma$ be the roots of $f$. The Berlekamp discriminant is \begin{equation*} B=\frac{\alpha\beta}{\alpha^2+\beta^2}+\frac{\beta\gamma}{\beta^2+\gamma^2}+\frac{\gamma\alpha}{\gamma^2+\alpha^2}. \end{equation*} If I computed correctly, then \begin{equation*} B=\frac{p^3+q^2}{q^2}. \end{equation*} Now the Galois group of $f$ over $K$ contains a transposition if and only if $B$ is not in the value set of $x^2+x$ on $K$. As we have at least one root of $f$ in $K$, this is equivalent to having exactly one root of $f$ in $K$.
Let $K$, the field generated by the coefficients of $f$, have $t=2^n$ elements. If $p=q=0$, $f$ has a triple root. If $q$ is not $0$, then $f$ is separable. Assume $q$ not $0$. If $p=0$ and n odd, then $f$ has exactly one root in $K$. If $p=0$ and $n$ even then $f$ has $0$ or $3$ roots in $K$. Assume now that $f$ has at least 1 root in $K$. Let $tr$ be the trace of $K$ over the prime field. Put $B=1+\frac{p^3}{q^2}$. Then $f$ has $3$ roots iff $tr(B)=0$.
The question if $f$ has no roots (i.e. $f$ irreducible) can be done by Berlekamp's algorithm.
