The Takagi existence theorem, the central theorem of classical class field theory, establishes a one-to-one, inclusion-reversing correspondence between the finite abelian extensions of a number field K and its generalized ideal class groups, proving that a number field always has enough abelian extensions for such a correspondence to exist. Teiji Takagi developed the theorem while working in Japan during the First World War and presented it at the 1920 International Congress of Mathematicians; at David Hilbert's request the paper appeared in Mathematische Annalen in 1925, where it unified earlier partial results, including Philipp Furtwangler's 1907 proof of the existence of the Hilbert class field, and set off the development of classical class field theory through the 1920s. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K and the generalized ideal class groups defined via a modulus of K. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Takagi existence theorem (Wikipedia)
Introduction, sentence 1Quote, Introduction, sentence 1
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
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