Splitting fields and general differential Galois theory

TL;DR

This paper introduces an algebraic method for developing a comprehensive Galois theory applicable to nonlinear systems of partial differential equations, avoiding model theory and focusing on prime differential ideals.

Contribution

It presents a novel algebraic approach to differential Galois theory that constructs differential closures and establishes Galois correspondence without model-theoretic results.

Findings

Constructed differential closure using algebraic techniques.

Established Galois correspondence for normal extensions.

Provided a framework for nonlinear PDE systems analysis.

Abstract

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..