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
is a principal ideal αOL, for OL 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.
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
is a principal ideal αOL, for OL 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.