The <Hilbert class field> $\widetilde K$ 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 map>
$$
\boxed{\operatorname{Cl}(K)\cong\operatorname{Gal}(\widetilde K/K),}
$$
so $[\widetilde K:K]$ is the <class number> of $K$.

Solution

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

An abelian extension is a <Galois extension> whose <Galois group> is an <abelian group>.

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