 Hilbert class field

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 givesAn 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.