Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 1 / a / Solution

A modulus of a number field is a formal productwhere ${\mathfrak{m}}_0={\prod }_{\mathfrak{p}}{\mathfrak{p}}^{n_{\mathfrak{p}}}$ is a nonzero integral ideal and ${\mathfrak{m}}_{\infty }$ is a product of distinct real embeddings of $K$. Only finitely many $n_{\mathfrak{p}}$ are nonzero; complex places do not occur.