nLab regulator of a number field

Idea

Given a number field $K$, with ring of integers $\mathcal{O}_K$, then the regulator $Reg_K$ is a number which measures 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 regulator 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 effectively 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 to the product of the regulator and the class number of $K$

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

Related concepts

• Dirichlet's unit theorem

• higher regulator

References

• Wikipedia, Dirichlet’s unit theorem – The regulator