A short note on groups in separably closed valued Fields

TL;DR

This paper demonstrates that groups with definable generics in separably closed valued fields of finite imperfection degree can be embedded into groups definable in their algebraic closure, revealing structural insights.

Contribution

It introduces a method to embed such groups into algebraically definable groups, advancing understanding of their algebraic and model-theoretic properties.

Findings

Groups with definable generics can be embedded into algebraic groups.

Embedding preserves key structural properties.

Provides a new perspective on the model theory of valued fields.

Abstract

In this note we show that groups with definable generics in a separably closed valued of finite imperfection degree can be embedded into groups definable in their algebraic closure.