# Constructing Number Field Isomorphisms from *-Isomorphisms of Certain Crossed Product C*-Algebras

## TL;DR

This paper shows how certain algebraic structures can reveal deep connections between number fields and their isomorphism properties.

## Contribution

A new method is introduced to construct number field isomorphisms from *-isomorphisms of specific C*-algebras.

## Key findings

- Crossed product C*-algebras associated with number fields are rigid under *-isomorphisms.
- An analogue of the Neukirch–Uchida theorem is proven using topological full groups.
- New discrete groups are identified that uniquely characterize number fields.

## Abstract

We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch–Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.