Fields With Finitely Many Non-Commutative Division Algebras Over Them

TL;DR

This paper classifies fields with finitely many finite non-commutative division algebras, introducing the concept of anti-closure and exploring fields with linear lattices of finite extensions.

Contribution

It provides a classification of such fields and introduces new concepts like anti-closure to understand their algebraic structure.

Findings

Classification of fields with finitely many non-commutative division algebras

Introduction of the anti-closure notion

Comments on fields with linear lattice of finite extensions

Abstract

We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear lattice of finite field extensions over them.