mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 cre ...

, a branch of

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...

, says that extending ideals gives a mapping on the class group of an

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

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

''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 considered ...

''I'' of the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

of ''K'', if ''L'' is the Hilbert class field 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 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 zer ...

(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