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, is the statement that 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. In other terms, extending ideals gives a mapping on the class group of K, to the class group of L, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called principalization, or sometimes capitulation. It was conjectured by David Hilbert, and was the last remaining aspect of his programme on class fields to be completed, around 1930.
- Furtwängler, Philipp (1929). "Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper". Abh. Math. Sem. Hamburg 7: 14–36. 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. 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 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. pp. 120–122. ISBN 0-387-90424-7. Zbl 0423.12016.