Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 4 / f / Solution

The Kronecker–Weber theorem says that every finite abelian extension of $\mathbb{Q}$ lies in some $\mathbb{Q}({\zeta }_m)$.

Indeed, global class field theory assigns to a finite abelian $L/\mathbb{Q}$ the open norm group of an abelian extension $N_{L/\mathbb{Q}}{C}_L$. It contains a congruence subgroup $C_{\mathbb{Q}}(m\infty )$. By part (e), this is the norm group of $\mathbb{Q}({\zeta }_m)$. The inclusion-reversing class-field correspondence therefore gives