Un Crit{\`E}Re Simple

TL;DR

This paper presents a criterion for simplicity in certain theories of fields with operators, based on mimicking proofs from difference field theory, linking simplicity to the control of definable and algebraic closures by a stable theory.

Contribution

It introduces a new criterion for simplicity in theories of fields with operators, extending ideas from difference field theory.

Findings

Provides a criterion for simplicity in theories of fields with operators.

Shows that a complete theory is simple if its closures are governed by a stable theory.

Links the simplicity of a theory to the stability of its definable and algebraic closures.

Abstract

In this short note, we mimic the proof of the simplicity of the theory ACFA of generic difference fields in order to provide a criterion, valid for certain theories of pure fields and fields equipped with operators, which shows that a complete theory is simple whenever its definable and algebraic closures are controlled by an underlying stable theory.