# 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. 1.

$h_{K}$ is the class number, the number of elements in the ideal class group of $K$.

2. 2.

$\operatorname{Reg}_{K}$ is the regulator of $K$.

3. 3.

$\omega_{K}$ is the number of roots of unity contained in $K$.

4. 4.

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

Then:

###### Theorem 1 (Class Number Formula).

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

 $\lim_{s\to 1}(s-1)\zeta_{K}(s)=\frac{2^{r_{1}}\cdot(2\pi)^{r_{2}}\cdot h_{K}% \cdot\operatorname{Reg}_{K}}{\omega_{K}\cdot\sqrt{\mid D_{K}\mid}}$

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