Differential Galois theory. I
type: Item
instance of: scholarly article
title: Differential Galois theory. I
zbMATH Open document ID: 0916.03028
published in: Illinois Journal of Mathematics
publication date: 1999-07-06T00:00:00Z
review text: This paper proposes a generalization of Kolchin's Galois theory of differential fields. In the Kolchin theory, the Galois groups correspond to algebraic groups over the subfield of constants; moreover every algebraic group can arise in this way. In this paper, the constants are replaced by an arbitrary differential algebraic set \(X\). Accordingly, \(X\)-strongly normal differential extensions are introduced. In detail, if \(F < K\) are differential fields and \(X\) is an \(F\)-definable set, \(K\) is called an \(X\)-strongly normal extension of \(F\) if \(X(F) = X(\overline{F})\) (where \(\overline{F}\) denotes a differential closure of \(F\)), \(\text{dcl}(X \cap F) \cap K = F\), \(K\) is finitely generated over \(F\) and, for any \(F\)-isomorphism \(f\) of \(K\) into the monster model, \(f(K) \subseteq K(X)\). \(K\) is called a generalized strongly normal extension of \(F\) if \(K\) is an \(X\)-strongly normal extension for some \(X\). Hence the Kolchin strongly normal differential extensions are generalized strongly normal (when \(X\) is the constant subfield). In the generalized setting, every Galois group is shown to be canonically isomorphic to a finite-dimensional differential algebraic group, unique up to \(F\)-definable isomorphism. It is also proved that, for \(K\) an \(X\)-strongly normal extension of \(F\) with Galois group \(G\), the Galois correspondence connects intermediate differential fields \(L\) between \(F\) and \(K\) with algebraic subgroups of \(G\). Moreover any finite-dimensional algebraic group arises as the Galois group of some generalized strongly normal extension \(F<K\) for suitable \(F\) and \(K\). The author observes that there are finite-dimensional differential algebraic groups which are not isomorphic (as differential algebraic groups) to algebraic groups over the constants; consequently, generalized strongly normal extensions actually enlarge Kolchin strongly normal extension. The paper provides also an analogue of Bialynicki-Bir…
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 1262683
zbMATH Keywords: differential field
zbMATH Keywords: Kolchin strongly normal extension
zbMATH Keywords: generalized strongly normal extension
zbMATH Keywords: finite-dimensional differential algebraic group
zbMATH Keywords: generalization of Kolchin's Galois theory of differential fields
author: Anand Pillay
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1262683