An idelic view of ideals

TL;DR

This paper explores the structure of ideles and adeles in algebraic number theory, providing foundational concepts, constructions, and applications such as proofs of Dirichlet's S-unit theorem and class group finiteness.

Contribution

It introduces a topological perspective on ideles and adeles, linking them to ideal class groups and offering alternative proofs of key theorems in number theory.

Findings

Ideles modulo $k^*$ surject onto the ideal class group

Compactness of $C_S^0$ leads to alternative proofs of Dirichlet's S-unit theorem

Finiteness of the ideal class group is established through topological methods

Abstract

Ideles and adeles can be viewed as a generalization of Minkowski theory, in which embedding of a number field to the Cartesian product of its completions at the archimedean valuation is generalized to an embedding of the Cartesian product of all its completions with some restriction. This paper introduces the basic notions of point-set topology and builds the real numbers from the rational numbers. Then we review concepts from local fields that will lead to the product formula and the approximation theorem. We, then, construct adeles and ideles. The ideles modulo $k^*$ maps surjectively to the ideal class group, and the compactness of $C_S^0$ will give rise to an alternative proof to the Dirichlet's S-unit theorem and the finiteness of ideal class group. The paper assumes that the reader is familiar with Dedekind domain, principal ideal domain, unique factorization of Dedekind domain, and field norm and traces. The first few parts of Chapter 1 in Neukirch's Algebraic Number Theory will be sufficient for the paper.