An algebraic proof of Brauer's class number formula

TL;DR

The paper addresses Brauer's class number formula by constructing an entirely algebraic proof based on regulator constants attached to Brauer relations. By leveraging the Tate exact sequences for $S$-units and class groups (Ritter–Weiss), the author connects regulator constants of units to those of class groups, and then to the ambiguous class number formula, ensuring all cohomological defects cancel. The main result shows that for any Brauer relation $\Theta$, $\prod_{H\le G}(h^S_{K^H})^{n_H}=\prod_{H\le G}\left(|\mu_{K^H}|/R^S_{K^H}\right)^{n_H}$, providing a general, torsion-aware algebraic derivation of Brauer–Kuroda-type relations. This framework avoids analytic methods and clarifies the role of regulator constants in expressing class numbers through roots of unity and regulators across subfields. The approach extends known special cases to arbitrary finite groups and invariant prime sets, offering a unifying algebraic perspective on Brauer’s formula.

Abstract

We show how, starting from the Tate exact sequence for units of Ritter and Weiss, one can obtain an algebraic proof of Brauer's class number formula, using the formalism of regulator constants.

Theorems & Definitions (50)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Example 2.8
• Example 2.9
• Remark 2.10