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.

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

The principal ideal theorem was conjectured by David Hilbert (1902), and was the last remaining aspect of his program on class fields to be completed, in 1929.

Emil Artin (1927, 1929) reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial. This result was proved by Philipp Furtwängler (1929).

References

• Artin, Emil (1927), "Beweis des allgemeinen Reziprozitätsgesetzes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5 (1): 353–363, doi:10.1007/BF02952531, S2CID 123050778
• 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, S2CID 121475651
• Furtwängler, Philipp (1929). "Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 7: 14–36. doi:10.1007/BF02941157. JFM 55.0699.02. S2CID 123544263.
• 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.
• Hilbert, David (1902) [1898], "Über die Theorie der relativ-Abel'schen Zahlkörper", Acta Mathematica, 26 (1): 99–131, doi:10.1007/BF02415486
• 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 by Greenberg, Marvin Jay. Springer-Verlag. pp. 120–122. ISBN 0-387-90424-7. Zbl 0423.12016.