Differential Galois theory. III: Some inverse problems
type: Item
instance of: scholarly article
title: Differential Galois theory. III: Some inverse problems
zbMATH Open document ID: 0927.03065
author: David Marker
author: Anand Pillay
published in: Illinois Journal of Mathematics
publication date: 1999-04-12T00:00:00Z
review text: [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.
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 1043159
zbMATH Keywords: differential field
zbMATH Keywords: differential Galois theory
zbMATH Keywords: differential algebraic group
zbMATH Keywords: differential Galois extensions
zbMATH Keywords: Galois group
zbMATH Keywords: superstable differential fields
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1043159