Differential Galois theory. III: Some inverse problems
[For Part I see ibid. 42, No. 4, 678-699 (1998; Zbl 0916.03028). Part II is reviewed above.] In Part I, the author developed a theory of differential Galois extensions, generalizing Kolchin's theory of strongly normal extensions. It was shown that arbitrary finite-dimensional differential algebraic groups can arise as differential Galois groups for this new theory; hence, as there do exist finite-dimensional differential algebraic groups which are not isomorphic to algebraic groups in the constants, this new approach properly extends Kolchin's theory. This paper is interested in the inverse problem within the new theory, and shows in particular the following result. Let \(F\) be a differential field such that the field of constants \(C(F)\) of \(F\) has infinite transcendence degree and \(F\) is the algebraic closure of a differential field finitely generated over \(C(F)\). Let \(A\) be either a simple abelian variety or a 1-dimensional algebraic torus over \(F\), \(G\) be its smallest differential algebraic subgroup. Then \(F\) has a differential Galois extension with Galois group \(G\). The proof uses model theoretic methods and some results of Buium and Cassidy on differential algebraic groups. The authors provide also an application related to the problem of finding superstable differential fields which are not differentially closed. In fact, they prove that, for a field of constants \(F\) of infinite transcendence degree, the strongly normal closure of \(F\) (i.e. the smallest algebraically closed differential extension in the differential closure \(\widehat{F}\) of \(F\) with no proper strongly normal extensions) cannot be superstable.
scientific article; zbMATH DE number 1043159