Splitting of differential quaternion algebras

TL;DR

This paper investigates the conditions under which differential quaternion algebras can be split by extensions, using Riccati equations to establish bounds on the transcendence degree of such splitting fields.

Contribution

It introduces bounds on the transcendence degree of splitting fields for differential quaternion algebras using Riccati equations, expanding understanding beyond algebraic extensions.

Findings

Differential quaternion algebras may not be split over algebraic extensions.

Solutions to Riccati equations provide bounds on splitting field transcendence degree.

Differential splitting fields can have higher transcendence degree than algebraic ones.

Abstract

We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field $k$ is always split by a quadratic extension of $k$. However, a differential quaternion algebra need not be split over any algebraic extension of $k$. We use solutions of certain Riccati equations to provide bounds on the transcendence degree of splitting fields of a differential quaternion algebra.