principal ideal theorem
{{short description|Theorem in class field theory on mappings induced by extending ideals}}
{{about|the Hauptidealsatz of class field theory|the theorem about Noetherian rings| 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
:
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 {{harvs|txt|first=David|last= Hilbert|authorlink=David Hilbert|year=1902}}, and was the last remaining aspect of his program on class fields to be completed, in 1929.
{{harvs|txt|first=Emil|last= Artin|year1=1927|year2=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
- {{citation|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
|year= 1927|volume= 5|issue =1|pages= 353–363
|title=Beweis des allgemeinen Reziprozitätsgesetzes
|first=Emil|last= Artin|doi=10.1007/BF02952531|s2cid= 123050778}}
- {{citation|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
|year= 1929|volume= 7|issue= 1|pages= 46–51
|title=Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz
|first=Emil|last= Artin|doi=10.1007/BF02941159|s2cid= 121475651}}
- {{cite journal | first=Philipp | last=Furtwängler | author-link=Philipp Furtwängler | title=Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=7 | year=1929 | issue=1 | pages=14–36 | doi=10.1007/BF02941157 | jfm=55.0699.02 | s2cid=123544263 }}
- {{cite book | last=Gras | first=Georges | title=Class field theory. From theory to practice | series=Springer Monographs in Mathematics | location=Berlin | publisher=Springer-Verlag | year=2003 | isbn=3-540-44133-6 | zbl=1019.11032 }}
- {{citation|journal=Acta Mathematica
|year=1902|orig-year=1898|volume =26|issue =1|pages= 99–131
|title=Über die Theorie der relativ-Abel'schen Zahlkörper
|first=David |last=Hilbert|doi=10.1007/BF02415486|doi-access=free}}
- {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=Springer-Verlag | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st | page=104 }}
- {{cite book | last=Serre | first=Jean-Pierre | author-link=Jean-Pierre Serre | title=Local Fields | translator-first1=Marvin Jay|translator-last1=Greenberg|translator-link1=Marvin Jay Greenberg | series=Graduate Texts in Mathematics | volume=67 | publisher=Springer-Verlag | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=120–122 }}