Prime Spectrum of the Ring of Adeles of a Number Field

TL;DR

This paper investigates the prime spectrum of the ring of adeles for a number field, exploring its algebraic and topological properties and their behavior under field extensions, with implications beyond number fields.

Contribution

It provides a detailed description of the prime spectrum of the adele ring and analyzes its algebraic and topological properties, a less-explored aspect in the literature.

Findings

Characterization of prime ideals in the adele ring

Analysis of topological properties of prime ideals

Behavior of prime ideals under field extensions

Abstract

Much is known about the adele ring of an algebraic number field from the perspective of Harmonic Analysis and Class Field Theory. However, its ring-theoretical aspects are often ignored. Here we present a description of the prime spectrum of this ring and study some of the algebraic and topological properties of these prime ideals. We also study how they behave under separable extensions of the base field and give an indication of how this study can be applied in adele rings not of number fields.