Hilbert class field characterization

A finite abelian extension $E/K$ of number fields lies in the ordinary Hilbert class field exactly when it is unramified at every finite prime and every real place of $K$ splits completely in $E$. Its Galois group is then a quotient of the ordinary ideal class group through the Artin map.

This gives a useful recognition test: construct such an extension and show that its degree equals the class number $h_K$. The inclusion $E\subseteq H_K$ and equality of degrees then force $E=H_K$. Finite-prime unramifiedness alone gives the corresponding narrow-class-field condition when real places may become complex.