 Solution

Regard $f,g$ as polynomials in $Y$ over $\mathbb{C}(X)$. Their coprimality and the preceding argument let Euclid's algorithm producewith coefficients in $\mathbb{C}(X)$. Clearing denominators gives a nonzero $h(X)\in (f,g)$, so $I\cap \mathbb{C}[X]\ne 0$. Interchanging $X,Y$ gives a nonzero $k(Y)\in I$.

In the quotient, $h(X)=k(Y)=0$. Reducing powers by these two univariate relations shows that the finitely many monomialsspan. Therefore $\mathbb{C}[X,Y]/I$ is finite-dimensional.