ideal classes form an abelian group
Let K be a number field, and let 𝒞 be the set of ideal classes of K, with multiplication ⋅ defined by
[𝔞]⋅[𝔟]=[𝔞𝔟] |
where 𝔞,𝔟 are ideals of 𝒪K.
We shall check the group properties:
-
1.
Associativity: [𝔞]⋅([𝔟]⋅[𝔠])=[𝔞]⋅[𝔟𝔠]=[𝔞(𝔟𝔠)]=[𝔞𝔟𝔠]=[(𝔞𝔟)𝔠]=[𝔞𝔟]⋅[𝔠]=([𝔞]⋅[𝔟])⋅[𝔠]
-
2.
Identity element
: [𝒪K]⋅[𝔟]=[𝔟]=[𝔟]⋅[𝒪K].
- 3.
Then 𝒞 is a group under the operation ⋅.
It is abelian since [𝔞][𝔟]=[𝔞𝔟]=[𝔟𝔞]=[𝔟][𝔞].
This is group is called the ideal class group of K. The ideal class group is one of the principal objects of algebraic number theory. In particular, for an arbitrary number field K, very little is known about the size of this group, called the class number of K. 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 |