The principal ideal theorem, in class field theory, states that extending the ideals of an algebraic number field to its Hilbert class field sends every ideal class to the class of a principal ideal. David Hilbert conjectured the result in 1902, Emil Artin reduced it between 1927 and 1929 to a question about finite abelian groups and transfers between them, and Philipp Furtwangler completed the proof in 1929, closing the last open part of Hilbert's program on class fields. The theorem is a foundational instance of principalization, the phenomenon by which non-principal ideals become principal once lifted into a suitable extension field. 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
StatementExtending ideals of an algebraic number field to its Hilbert class field sends every ideal class to the class of a principal ideal. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
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.
In Branch
Source Principal ideal theorem (Wikipedia)
Sources
1. Principal ideal theorem (Wikipedia)
Introduction
extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal
History
was the last remaining aspect of his program on class fields to be completed, in 1929
In Branch: Algebraic Number Theory, Lead sentence
ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class
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