On the Galois theory of difference fields (Q1163058)

By adapting ideas from Kolchin's study of differential fields, it is possible to develop a Galois theory for difference fields. The theory applies to difference field extensions \(M/F\) of characteristic zero with \(F\) algebraically closed in \(M\) and satisfying a condition on constants called strong normality. If \(\sigma\) is a difference isomorphism of \(M/F\), let \(C_\sigma\) be the field of constants of \(M\langle \sigma M\rangle\). Then \(M/F\) is a strongly normal extension if \(M\langle C_\sigma\rangle = M\langle \sigma M\rangle = \sigma M\langle C_\sigma\rangle\) for all \(\sigma\). Given this condition and a technical condition on constants (which can be removed) then there is a connected algebraic group, \(G\), defined over the constants of \(F\) and a Galois correspondence between the connected subgroups of \(G\) and the relatively closed intermediate fields of \(M/F\). In the second section of the paper a class of strongly normal extensions is constructed having the property that the Galois group of each member of the class is not a linear group. Therefore, the present theory generalizes that given by Charles Franke in his study of solution fields of linear homogeneous difference equations. Finally, genus 1 extensions are characterized by showing that the Galois groups are elliptic curves.