ideal class
Let be a number field. Let and be ideals in (the ring of algebraic integers of ). Define a relation on the ideals of in the following way: write if there exist nonzero elements and of such that .
The relation is an equivalence relation, and the equivalence classes under are known as ideal classes.
The number of equivalence classes, denoted by or , is called the class number of .
Note that the set of ideals of any ring forms an abelian semigroup with the product of ideals as the semigroup operation. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of in the following way.
Let , be ideals of . Denote the ideal classes of which and are representatives by and respectively. Then define by
Let . With the above definition of multiplication, is an abelian group, called the ideal class group (or frequently just the class group) of .
Note that the ideal class group of is simply the quotient group of the ideal group of by the subgroup of principal fractional ideals.
Title | ideal class |
Canonical name | IdealClass |
Date of creation | 2013-03-22 12:36:42 |
Last modified on | 2013-03-22 12:36:42 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 22 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 11R29 |
Synonym | ideal classes |
Related topic | ExistenceOfHilbertClassField |
Related topic | FractionalIdeal |
Related topic | NumberField |
Related topic | UnramifiedExtensionsAndClassNumberDivisibility |
Related topic | ClassNumberDivisibilityInExtensions |
Related topic | PushDownTheoremOnClassNumbers |
Related topic | MinkowskisConstant |
Related topic | ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivis |
Defines | class number |
Defines | ideal class group |
Defines | class group |