A remark on Chevalley's ambiguous class number formulas

TL;DR

This paper highlights that Chevalley's ambiguous class number formula can be directly derived from fundamental theorems like the Hasse norm theorem and local-global norm index theorems for cyclic extensions.

Contribution

It provides a simple, direct proof of Chevalley's ambiguous class number formula using well-known theorems, clarifying its foundational basis.

Findings

Chevalley's formula follows from the Hasse norm theorem.

The proof relies on local and global norm index theorems.

The approach simplifies understanding of the ambiguous class number formula.

Abstract

In this note we remark that Chevalley's ambiguous class number formula is an immediate consequence of the Hasse norm theorem, the local and global norm index theorems for cyclic extensions.