Differential Galois Theory, Volume 46, Number 9

D ifferential Galois theory, like the more familiar Galois theory of polynomial equations on which it is modeled, aims to understand solving differential equations by exploiting the symmetry group of the field generated by a complete set of solutions to a given equation. The subject was invented in the late nineteenth century, and by the middle of the twentieth had been recast in modern rigorous form. But despite being an active subject of contemporary research, and an important tool in applications, and despite the availability of texts and courses on the subject, the basic results of the subject seem not to be widely known. This article is intended to provide a gateway to those results. The late nineteenth-century work was done by Picard and Vessiot. The modern rigorous form of the subject is due to E. Kolchin. The basic theorems of differential Galois theory seem by now to have entered the public domain, however, and are presented here without reference or attribution. It is safe to assume that they have their origins in Kolchin’s work; none should be thought of as work of the present author. Before starting the discussion of differential Galois theory, we review the fundamental concepts of ordinary, here termed “polynomial”, Galois theory. Both Galois theories involve an extension of fields, and each has a Fundamental Theorem. Making use of Galois theory in concrete situations requires being able to compute groups of automorphisms, and this and the inverse problem remain active areas of research. The corresponding problems of differential Galois theory are the ultimate subjects of this article.