Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 10E / b

Reduce coefficients modulo $I$:This is a surjective ring homomorphism and its kernel consists exactly of polynomials all of whose coefficients lie in $I$, namely $I[X]$. The first isomorphism theorem gives the coefficientwise quotient of a polynomial ringIf $I$ is prime, then $R/I$ is an integral domain, so $(R/I)[X]$ is an integral domain. The quotient criterion therefore shows that $I[X]$ is prime in $R[X]$.

Take $R=\mathbb{Z}$ and $I=(2)$. The ideal $I$ is maximal, butis not a field, since $X$ is nonzero and not invertible. Therefore