# Contents

## Idea

Given a number field $K$, with ring of integers $\mathcal{O}_K$, then the regular $Reg_K$ is a number which measure the size of the group of units $GL_1(\mathcal{O}_K)$.

Since the determinant constitutes a group homomorphism $K_1 \to GL_1$ from first algebraic K-theory to the group of units, the reguöator may also be thought of as extracting information on the first algebraic K-theory group. This is the perspective taken in the generalization to higher regulators (Beilinson regulators) which are effectivlely Chern characters for algebraic K-theory.

## Properties

### Relation to zeta function and class number formula

The Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional the the product of the regulator with the class number of $K$

$\underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.$

## References

Created on August 25, 2014 at 23:25:35. See the history of this page for a list of all contributions to it.