Principal ideal theorem

This article is about the Hauptidealsatz of class field theory. For the theorem about Noetherian rings, see Krull's principal ideal theorem.

In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that 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. The phenomenon has also been called principalization, or sometimes capitulation.