In mathematics, the principal ideal theorem of

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

, a branch of algebraic number theory, says that extending ideals gives a mapping on the

class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...

of an algebraic number field to the class group of its

Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the Maximal abelian extension, maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' ...

, 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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

''I'' of the ring of integers of ''K'', if ''L'' is the

of ''K'', then :

IO_L\

is a

principal ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...

α''O''_''L'', for ''O''_''L'' the ring of integers of ''L'' and some element α in it.

History

The principal ideal theorem was conjectured by , and was the last remaining aspect of his program on class fields to be completed, in 1929. reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the

transfer Transfer may refer to: Arts and media * ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović * ''Transfer'' (1966 film), a short film * ''Transfer'' (journal), in management studies ...

from a finite group to its derived subgroup is trivial. This result was proved by

Philipp Furtwängler Friederich Pius Philipp Furtwängler (April 21, 1869 – May 19, 1940) was a German number theorist. Biography Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (''Zur Theorie der in Linearfaktoren ze ...

(1929).

References

* * * * * * *{{cite book , last=Serre , first=Jean-Pierre , author-link=Jean-Pierre Serre , title=

Local Fields ''Corps Locaux'' by Jean-Pierre Serre, originally published in 1962 and translated into English as ''Local Fields'' by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, ...

, translator-first1=Marvin Jay, translator-last1=Greenberg, translator-link1=Marvin Jay Greenberg , series=

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...

, volume=67 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

, year=1979 , isbn=0-387-90424-7 , zbl=0423.12016 , pages=120–122 Ideals (ring theory) Group theory Homological algebra Theorems in algebraic number theory