class number formula

Let $K$ be a number field with $[K:\mathbb{Q}]=n=r_1+2r_2$, where $r_1$ denotes the number of real embeddings of $K$, and $2r_2$ is the number of complex embeddings of $K$. Let

$${\zeta }_K(s)$$

be the Dedekind zeta function of $K$. Also define the following invariants:

1. $h_K$ is the class number, the number of elements in the ideal class group of $K$.

2. ${\mathrm{Reg}}_K$ is the regulator of $K$.

3. ${\omega }_K$ is the number of roots of unity contained in $K$.

4. $D_K$ is the discriminant of the extension $K/\mathbb{Q}$.

Then:

Theorem 1 (Class Number Formula).

The Dedekind zeta function of $K$, ${\zeta }_K\mathit{}\mathrm{(}s\mathrm{)}$ converges absolutely for $\mathrm{\Re }\mathit{}\mathrm{(}s\mathrm{)}\mathrm{>}\mathrm{1}$ and extends to a meromorphic function defined for $\mathrm{\Re }\mathit{}\mathrm{(}s\mathrm{)}\mathrm{>}\mathrm{1}\mathrm{-}\frac{\mathrm{1}}{n}$ with only one simple pole at $s\mathrm{=}\mathrm{1}$. Moreover:

$$\underset{s\to 1}{lim}(s-1){\zeta }_K(s)=\frac{2^{r_1}\cdot {(2\pi )}^{r_2}\cdot h_K\cdot {\mathrm{Reg}}_K}{{\omega }_K\cdot \sqrt{\mid D_K\mid }}$$

Note: This is the most general “class number formula”. In particular cases, for example when $K$ is a cyclotomic extension of $\mathbb{Q}$, 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