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.
Read more about Takagi Existence Theorem: Formulation, A Well-defined Correspondence, Earlier Work, History
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