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

If the embedding with unit modulus is real, it sends $\alpha$ to $\pm 1$; irreducibility of the minimal polynomial then forces $\alpha =\pm 1$. Otherwise $\overline{{\sigma }_j(\alpha )}=1/{\sigma }_j(\alpha )$ is another conjugate. Hence ${\alpha }^{-1}$ is conjugate to $\alpha$, so their monic minimal polynomials coincide. The minimal polynomial is therefore reciprocal up to its constant sign, and its constant term has absolute value one. Taking the product of all conjugates gives

If $\alpha \in {\mathcal{O}}_K$, its norm is an integer. The preceding result gives norm $\pm 1$. The characteristic polynomial of multiplication by $\alpha$ then expresses ${\alpha }^{-1}$ as an integral polynomial in $\alpha$, so ${\alpha }^{-1}\in {\mathcal{O}}_K$. Therefore

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.