Division algebras with the same maximal subfields

TL;DR

This paper surveys recent advances in characterizing finite-dimensional division algebras through their maximal subfields, explores generalizations, and extends the problem to algebraic groups, highlighting key results and applications.

Contribution

It provides a comprehensive overview of recent results and extends the problem to the context of algebraic groups, offering new perspectives and applications.

Findings

Characterization of division algebras by maximal subfields

Generalizations of the problem to broader algebraic structures

Applications to algebraic groups and related areas

Abstract

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.