Differential Galois theory. I
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-Birula's approach in the generalized setting, as well as an analogue of Kolchin's \(G\)-primitive extensions characterization of strongly normal extensions. Throughout the paper, model theory (in particular, model theory of differential fields) is largely used. The necessary model-theoretic notions and tools are shortly summarized in the introduction. The author underlines their role as an efficient language to develop and express the whole theory.
finite-dimensional differential algebraic group
generalization of Kolchin's Galois theory of differential fields
scientific article; zbMATH DE number 1262683