ideal classes form an abelian group
Let be a number field, and let be the set of ideal classes of , with multiplication defined by
where are ideals of .
We shall check the group properties:
- 1.
- 2.
- 3.
Then is a group under the operation .
It is abelian since .
This is group is called the ideal class group of . The ideal class group is one of the principal objects of algebraic number theory. In particular, for an arbitrary number field , very little is known about the size of this group, called the class number of . See the analytic class number formula.
Title | ideal classes form an abelian group |
Canonical name | IdealClassesFormAnAbelianGroup |
Date of creation | 2013-03-22 12:49:40 |
Last modified on | 2013-03-22 12:49:40 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 13 |
Author | mathcam (2727) |
Entry type | Theorem |
Classification | msc 11R04 |
Classification | msc 11R29 |
Related topic | NumberField |
Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |
Related topic | FractionalIdealOfCommutativeRing |
Defines | ideal class group |
Defines | class number |