Algebraic families of subfields in division rings

TL;DR

This paper explores the relationships between maximal subfields in division rings and their rational extensions, demonstrating how properties like Galois or purely inseparable extend generically, with applications to enveloping skewfields in positive characteristic.

Contribution

It establishes new connections between subfield properties in division rings and their extensions, and describes the structure of enveloping algebras in positive characteristic.

Findings

Existence of Galois and purely inseparable maximal subfields in enveloping skewfields

Properties like Galois extend generically between division rings and their extensions

Description of enveloping algebra of p-envelope as polynomial extension

Abstract

We describe relations between maximal subfields in a division ring and in its rational extensions. More precisely, we prove that properties such as being Galois or purely inseparable over the centre generically carry over from one to another. We provide an application to enveloping skewfields in positive characteristics. Namely, there always exist two maximal subfields of the enveloping skewfield of a solvable Lie algebra, such that one is Galois and the second purely inseparable of exponent 1 over the centre. This extends results of Schue in the restricted case. Along the way we provide a description of the enveloping algebra of the p-envelope of a Lie algebra as a polynomial extension of the smaller enveloping algebra.