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

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

Solved by gpt-5.6-sol high.