Takagi Existence Theorem - A Well-defined Correspondence

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.