{{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
:<math>IO_L\ </math>
is a principal ideal α''O''<sub>''L''</sub>, for ''O''<sub>''L''</sub> 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 }}
Category:Ideals (ring theory) Category:Group theory Category:Homological algebra Category:Theorems in algebraic number theory