Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 123 / 3 / ii / Solution

The Hilbert class field $\tilde{K}$ is the maximal everywhere unramified abelian extension of the number field $K$, with complete splitting at real places if infinite places are included. Global class field theory gives the canonical Artin reciprocity mapso $[\tilde{K}:K]$ is the class number of $K$.