A uniform proof of the finiteness of the class group of a global field

TL;DR

This paper provides a unified proof that the class group of rings of integers in global fields is finite, establishing a fundamental property of algebraic number theory.

Contribution

It introduces a broad class of Dedekind domains encompassing rings of integers of global fields and proves their class groups are finite, unifying previous results.

Findings

All rings in the defined class have finite ideal class group.

The class coincides exactly with rings of integers of global fields.

The proof is uniform across the class.

Abstract

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of integers of global fields.