Hilbert class field

Hilbert class field

In algebraic number theory, the Hilbert class field "E" of a number field "K" is the maximal abelian unramified extension of "K". Its degree over "K" equals the class number of "K" and the Galois group of "E" over "K" is canonically isomorphic to the ideal class group of "K" using Frobenius elements for prime ideals in "K".

Note that in this context, the Hilbert class field of "K" is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of "K". That is, every real embedding of "K" extends to a real embedding of "E" (rather than to a complex embedding of "E"). As an example of why this is necessary, consider the real quadratic field "K" obtained by adjoining the square root of 3 to Q. This field has class number 1, but the extension "K"(i)/"K" is unramified at all prime ideals in "K", so "K" admits finite abelian extensions of degree greater than 1 in which all primes of "K" are unramified. This doesn't contradict the Hilbert class field of "K" being "K" itself: every proper finite abelian extension of "K" must ramify at some place, and in the extenson "K"(i)/"K" there is ramification at the archimedean places: the real embeddings of "K" extend to complex (rather than real) embeddings of "K"(i).

The existence of unique Hilbert class field for given number field "K" was conjectured by David Hilbert and proved by Philipp Furtwängler. The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of given field.

Additional properties

Furthermore, "E" satisfies the following:
*"E" is a finite Galois extension of "K" and ["E" :" K"] ="h""K", where "h""K" is the class number of "K".
*The ideal class group of "K" is isomorphic to the Galois group of "E" over "K".
*Every ideal of "O""K" is a principal ideal of the ring extension "O""E" (principal ideal theorem).
*Every prime ideal "P" of "O""K" decomposes into the product of "h""K"/"f" prime ideals in "O""E", where "f" is the order of ["P"] in the ideal class group of "O""K".

In fact, "E" is the unique field satisfying the first, second, and fourth properties.

References

----


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Class field theory — In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields. Most of the central results in this area were proved in the period between 1900 and 1950. The theory takes its name… …   Wikipedia

  • Non-abelian class field theory — In mathematics, non abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois… …   Wikipedia

  • Class formation — In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate. Contents 1 Definitions 2 Examples of class formations 3 The …   Wikipedia

  • Hilbert's twelfth problem — Hilbert s twelfth problem, of the 23 Hilbert s problems, is the extension of Kronecker Weber theorem on abelian extensions of the rational numbers, to any base number field. The classical theory of complex multiplication does this for any… …   Wikipedia

  • Hilbert's tenth problem — is the tenth on the list of Hilbert s problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it… …   Wikipedia

  • Hilbert's ninth problem — Hilbert s ninth problem, from the list of 23 Hilbert s problems (1900), asked to find the most general reciprocity law for the norm residues of l th order in a general algebraic number field, where l is a power of a prime. Progress made The… …   Wikipedia

  • David Hilbert — Hilbert redirects here. For other uses, see Hilbert (disambiguation). David Hilbert David Hilbert (1912) Born …   Wikipedia

  • Hilbert symbol — In mathematics, given a local field K, such as the fields of reals or p adic numbers, whose multiplicative group of non zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the… …   Wikipedia

  • Hilbert's Theorem 90 — In number theory, Hilbert s Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of number fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it tells us that if L / K is a cyclic… …   Wikipedia

  • Ideal class group — In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”