Multiplicative groups of division rings

TL;DR

This paper surveys recent developments in understanding the complex structure of the multiplicative groups of division rings, focusing on quaternion algebras, which are key examples of non-commutative division rings.

Contribution

It provides a comprehensive overview of recent progress in analyzing the multiplicative group structure of division rings, especially quaternion algebras.

Findings

Recent advances in the structure theory of quaternion algebra groups

Open problems and conjectures in the multiplicative group analysis

Connections between division ring properties and their multiplicative groups

Abstract

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.