Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 4 / d / ii / Solution

Conversely, let $A$ be a Noetherian UFD of Krull dimension one. Every nonzero prime is minimal among nonzero primes, and the forward argument in part (c) makes it principal. The zero ideal is principal as well, so part (b) shows that $A$ is a PID. A field has Krull dimension zero, hence $A$ is not a field.