In class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K (in a fixed algebraic closure of K) and the generalized ideal class groups defined via a modulus of K. It is called an existence theorem because a main burden of the proof is to show the existence of enough abelian extensions of K. Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents.
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.
The existence theorem is due to Takagi, who proved it in Japan during the isolated years of World War I.