# On Four-Dimensional Unital Division Algebras over Fields of Characteristic not 2

## TL;DR

This paper extends the classification of four-dimensional unital division algebras from finite fields to arbitrary fields of characteristic not 2, providing an explicit construction, isomorphism criteria, and automorphism group descriptions.

## Contribution

It generalizes previous finite field results to all fields of characteristic not 2, offering an exhaustive construction and classification of these division algebras.

## Key findings

- Explicit construction depending on quadratic field extensions and parameters
- Isomorphism criteria in terms of parameters
- Classification of automorphism groups for all such division algebras

## Abstract

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We generalize the approach of [2] towards all division algebras of the above specified type, but now admitting arbitrary fields k of characteristic not 2 as ground fields. For these division algebras we present an exhaustive construction that depends on a quadratic field extension of k and three parameters in k, and we derive an isomorphism criterion in terms of these parameters. As an application we classify, for o an ordered field in which every positive element is a square, all division o-algebras of the mentioned type, and in the finite field case we refine the Main Theorem of [2] to a classification even of the division algebras studied there. The category formed by the division k-algebras investigated here is a groupoid, whose structure we describe in a supplementary section in terms of a covering by group actions. In particular, we exhibit the automorphism groups for all division algebras in this groupoid.