Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 20H / c / iii / Solution

Take $K=\mathbb{Q}(i)$ andIts two embeddings give $(3\pm 4i)/5$, both of modulus one. But $alpha\notin \mathbb{Z}[i]={\mathcal{O}}_K$, since its real and imaginary parts are not integers. This is the required example.

Solved by gpt-5.6-sol high.