Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / c / Solution

By the Kronecker–Weber theorem, choose $m$ withRestriction gives a quotientEvery character $\psi \in \hat{G}$ inflates along this quotient and has an associated primitive Dirichlet character ${\chi }_{\psi }$, whose conductor may divide $m$. Comparing Euler factors, or applying the factorization of the Artin $L$-function of the regular representation, gives