Given a number field K, there exists a unique maximal unramified Abelian extension L of K which contains all other unramified Abelian extensions of K. This finite field extension L is called the Hilbert class field of K. By a theorem of class field theory, the Galois group G = Gal(L/K) is isomorphic to the class group of K and for every subgroup G' of G, there exists a unique unramified Abelian extension L' of K contained in L such that G' = Gal(L/L'). The degree [L:K] of L over K is equal to the class number of K.

Artin map | Artin symbol | class field | class field theory | class number