Given a number field , there exists a unique maximal unramified Abelian extension of which contains all other unramified Abelian extensions of . This finite field extension is called the Hilbert class field of . By a theorem of class field theory, the Galois group is isomorphic to the class group of and for every subgroup of , there exists a unique unramified Abelian extension of contained in such that .
The degree of over is equal to the class number of .