the same lines of the proof that the usual ideal class group is finite, well in advance of knowing these are Galois groups of finite abelian extensions of the number field. A well-defined correspondence Strictly speaking, the correspondence between finite abelian extensions of K and generalized ideal class groups is not quite one-to-one. Generalized ideal class groups defined relative to different moduli can give rise to the same abelian extension of K, and this is codified a priori in a somewhat complicated equivalence relation on generalized ideal class groups. In concrete terms, for abelian extensions L of the rational numbers, this corresponds to the fact that an abelian extension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic fields, and for each such cyclotomic overfield one obtains by Galois theory a subgroup of the Galois group corresponding to the same field L. In the idelic formulation of class field theory, one obtains a precise one-to-one correspondence between abelian extensions and appropriate groups of ideles, where equivalent generalized ideal class groups in the ideal-theoretic language correspond to the same group of ideles. Earlier work A special case of the existence theorem is when m = 1 and H = P1. In this case the generalized ideal class group is the ideal class group of K, and the existence theorem says there exists a unique abelian extension L/K with Galois group isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field. It was conjectured by David Hilbert to exist, and existence in this special case was proved by Furtwängler in 1907, before Takagi's general existence theorem. A further and special property of the Hilbert class field, not true of smaller abelian extensions of a number field, is that all ideals in a number field become principal in the Hilbert class field. It required Artin and Furtwängler to prove that principalization occurs. History The existence theorem is due to Takagi, who proved it in Japan during the isolated years of World War I. He presented it at the International Congress of Mathematicians in 1920, leading to the development of the classical theory of class field theory during the 1920s. At Hilbert's request, the paper was published in Mathematische Annalen in 1925. See also Class formation References Helmut Hasse, History of Class