Let $k$ be a field, and consider an irreducible polynomial $f \in k[x,y]$. Let $S(f)$ denote the singular points of $f$ (points that are simultaneously zero on $f$, the $x$-derivative of $f$, and the $y$-derivative of $f$.)

If $k$ is algebraically closed, then I can prove $S(f)$ is finite. Also, I can prove that if the field has characteristic $0$, then $S(f)$ is finite.

But what if the field has characteristic $p$ and is not algebraically closed? Is it true that $S(f)$ is finite?

I asked this question to my algebraic geometry professor last semester and stumped him! Hopefully one of you can think of a counterexample or proof.