Splitting fields of differential symbol algebras

TL;DR

This paper investigates differential symbol algebras of degree m, constructing explicit differential splitting fields and analyzing their properties, including bounds on degrees and maximal subfields, in characteristic not dividing m.

Contribution

It provides explicit constructions of differential splitting fields for symbol algebras and analyzes their algebraic and transcendence degrees, advancing understanding of their structure.

Findings

Explicit differential splitting fields constructed for certain derivations

Bounds established on algebraic and transcendence degrees of splitting fields

Analysis of maximal subfields that split differential symbol algebras

Abstract

For $m\geq 2$, we study derivations on symbol algebras of degree $m$ over fields with characteristic not dividing $m$. A differential central simple algebra over a field $k$ is split by a finitely generated extension of $k$. For certain derivations on symbol algebras, we provide explicit construction of differential splitting fields and give bounds on their algebraic and transcendence degrees. We further analyze maximal subfields that split certain differential symbol algebras.