Principal ideal theorem

This article is about the Hauptidealsatz of class field theory. You may be seeking Krull's principal ideal theorem, also known as Krull's Hauptidealsatz, in commutative algebra

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.

Formal statement

For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then

IO_{L}\

is a principal ideal αO_L, for O_L the ring of integers of L and some element α in it.

History

It was conjectured by David Hilbert, and was the last remaining aspect of his programme on class fields to be completed, around 1930.

The question was reduced to a piece of finite group theory by Emil Artin. That involved the transfer. The required result was proved by Philipp Furtwängler.

References

Artin, Emil (1927), "Beweis des allgemeinen Reziprozitätsgesetzes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5 (1): 353–363, doi:10.1007/BF02952531
Artin, Emil (1929), "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 7 (1): 46–51, doi:10.1007/BF02941159
Furtwängler, Philipp (1929). "Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper". Abh. Math. Sem. Hamburg. 7: 14–36. doi:10.1007/BF02941157. JFM 55.0699.02.
Gras, Georges (2003). Class field theory. From theory to practice. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-44133-6. Zbl 1019.11032.
Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. p. 104. ISBN 3-540-63003-1. Zbl 0819.11044.
Serre, Jean-Pierre (1979). Local fields. Graduate Texts in Mathematics. Vol. 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. pp. 120–122. ISBN 0-387-90424-7. Zbl 0423.12016.