# Equational theories of fields

## TL;DR

This paper investigates equational theories of fields, establishing equationality for various classes of algebraically closed and separably closed fields, and providing new proofs for known results in differential fields.

## Contribution

It proves the equationality of the theories of proper extensions of algebraically closed fields and separably closed fields, and offers a new proof for the equationality of differentially closed fields in positive characteristic.

## Key findings

- Equationality of proper extensions of algebraically closed fields.
- Equationality of separably closed fields of any imperfection degree.
- Alternative proof for the equationality of differentially closed fields in positive characteristic.

## Abstract

A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields of some fixed characteristic and of the theory of separably closed fields of arbitrary imperfection degree. Srour showed that the theory of differentially closed fields in positive characteristic is equational. We give also a different proof of his result.