= Hilbert class field characterization
{c}
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.
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