Embedding stable groups into algebraic groups

TL;DR

This paper proves that in stable theories of fields, every type-definable connected group can be embedded into an algebraic group, under certain conditions, providing a uniform framework for understanding definable closures.

Contribution

It extends embedding results to a broad class of stable field theories and offers general conditions for describing definable closures uniformly.

Findings

Every type-definable connected group embeds into an algebraic group.

Provides conditions for uniform description of definable closure in stable theories of fields.

Applicable to theories of separably closed and differentially closed fields.

Abstract

Adapting a proof of Bouscaren and Delon, we show that every type-definable connected group in a given stable theory of fields embeds into an algebraic group, under a condition on the definable closure. We also present general hypotheses which yield a uniform description of the definable closure in such theories of fields. The setting includes in particular the theories of separably closed fields of arbitrary degree of imperfection and differentially closed fields of arbitrary characteristic.