Let $\bar k$ be an algebraically closed field. Consider $f,g\in\bar k[x,y]$ such that $f$ and $g$ are co-prime. We want to show that the set of common roots of $f$ and $g$ is finite.
Here is my initial approach: we can only consider the case where $f$ is irreducible and $f$ does not divide $g$. Now, consider the image of $g$ in $\bar k[x,y]/(f)$. Since the zeroes of $g$ in $\bar k[x,y]$ correspond to the maximal ideals, images $(x-a,y-b)$, we want to prove that $g$ belongs to finitely many maximal ideals in $\bar k[x,y]$. Now how do we prove this?
I am aware of the proof using resultant. I am trying to find an algebraic proof.
