Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 9E (Groups, Rings and Modules)

A polynomial over a unique factorization domain is primitive when its coefficients have no common nonunit factor. If a prime divided every coefficient of a product of two primitive polynomials, reduction modulo that prime would turn the product of two nonzero polynomials over an integral domain into zero, impossible. This proves Gauss lemma for polynomials.

If positive-degree $f,g\in R[X]$ became non-coprime in $F[X]$, they would share a positive-degree primitive factor after clearing content. Gauss's lemma then makes that factor divide both in $R[X]$, contrary to coprimality. Hence they remain coprime over the fraction field.

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.