The common zeroes of two homogeneous polynomials $F$ and $G$ in $\mathbb P^2$ are finite when they have no common factors.
My thought on this which might (not) be helpful:
I know we may use Bezout's theorem, which requires one of two polynomials to have no singularities. I feel like we need to somehow use the condition that they have no common factors to make one of them not singular?
I wish the proof can be an algebraic one rather than a geometric one. However, any solution will be appreciated!