The Inverse Problem in the Galois Theory of Differential Fields

Abstract

In this paper we continue the study of the inverse problem in the Galois theory of differential field extensions which was begun in Kovacic [3]. (We shall refer to that paper as IP.) In Chapter I we develop the theory of the logarithmic derivative in the case of abstract group varieties (i.e. connected algebraic groups) and apply the theory to the inverse problem. We are able to show that the inverse problem for arbitrary abstract group varieties may be reduced to two problems: the inverse problem for abelian varieties and the inverse problem for connected linear groups. Also in this chapter we introduce an equivalence relation on the Lie algebra which differs slightly from that of IP. Except for the reduction noted above, Chapter I of the present paper closely parallels Chapter I of IP. In Chapter II we treat the inverse problem for abelian varieties. We are able to show the existence of a strongly normal extension (and indeed many non-isomorphic extensions) for any abelian variety over a differential field which either has finite transcendence degree over its field of constants, or is contained in the field of power series in one variable over constants and contains the rational functions. In Chapter III we discuss the inverse problem for a connected linear group. We essentially reduce the problem to the case in which the group is a power of a simple group. The author is indebted to Professor E. R. Kolchin for his many suggestions. We note that his work on G-primitives, where G is an abelian variety defined over the complex numbers, (Kolchin [1]) leads to a solution of the inverse problem for fields of functions of one variable. However, in this paper, we have chosen a different route to lead to this result.