= Hilbert class field
{c}
{title2=$H_K$}
{wiki}
The ordinary <Hilbert class field> $H_K$ of a <number field> $K$ is its maximal finite <abelian extension> unramified at every finite prime and in which every real place splits completely. <Artin reciprocity> gives
$$
\operatorname{Gal}(H_K/K)\cong\operatorname{Cl}(K),\qquad[H_K:K]=h_K.
$$
An unramified prime splits completely in $H_K$ exactly when its <ideal class> is trivial. For $K=\mathbb Q$, the <ring of integers> is the <principal ideal domain> $\mathbb Z$, so $H_K=\mathbb Q$.
The split-real condition fixes the ordinary convention. Allowing real places to become complex instead corresponds to the ray class <field> for the product of all real places, whose group is the <narrow ideal class group>. For a <totally imaginary number field>, there is no distinction at infinite places.
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