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class number formula (Theorem)

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

$\displaystyle \zeta_K(s)$
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. $ \operatorname{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(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:
$\displaystyle \lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot \operatorname{Reg}_K}{\omega_K \cdot \sqrt{\mid D_K \mid}}$

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.



"class number formula" is owned by alozano.
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See Also: functional equation of the Riemann zeta function, Dedekind zeta function, ideal class, regulator, discriminant, number field, topics on ideal class groups and discriminants

Also defines:  class number formula
Keywords:  class number, Dedekind zeta function
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Cross-references: cyclotomic extension, simple pole, function, meromorphic, converges absolutely, extension, discriminant, contained, roots of unity, regulator, ideal class group, class number, invariants, Dedekind zeta function, complex embeddings, real embeddings, number field
There are 6 references to this entry.

This is version 2 of class number formula, born on 2003-08-29, modified 2003-08-29.
Object id is 4664, canonical name is ClassNumberFormula.
Accessed 4121 times total.

Classification:
AMS MSC: 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
 11R42 (Number theory :: Algebraic number theory: global fields :: Zeta functions and $L$-functions of number fields)
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