class number formula
Let be a number field with , where denotes the number of real embeddings of , and is the number of complex embeddings of . Let
be the Dedekind zeta function of . Also define the following invariants:
-
1.
is the class number, the number of elements in the ideal class group of .
-
2.
is the regulator of .
-
3.
is the number of roots of unity contained in .
-
4.
is the discriminant of the extension .
Then:
Theorem 1 (Class Number Formula).
The Dedekind zeta function of , converges absolutely for and extends to a meromorphic function defined for with only one simple pole at . Moreover:
Note: This is the most general “class number formula”. In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas.
Title | class number formula |
Canonical name | ClassNumberFormula |
Date of creation | 2013-03-22 13:54:37 |
Last modified on | 2013-03-22 13:54:37 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11R29 |
Classification | msc 11R42 |
Related topic | FunctionalEquationOfTheRiemannZetaFunction |
Related topic | DedekindZetaFunction |
Related topic | IdealClass |
Related topic | Regulator |
Related topic | Discriminant |
Related topic | NumberField |
Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |
Defines | class number formula |