Questions on Skew Fields

Abstract

There are certain skew fields that arise quite naturally and about which we know very little at present; they include the skew fields of fractions of the Weyl algebras henceforth called the Weyl skew fields and written as D_n, which are closely linked to the skew fields of factions of the enveloping algebras of finite dimensional Lie algebras, and the skew fields of fractions of group rings of torsion free polycyclic by finite groups; all these may be subsumed in the class of skew fields that are iterated Ore or finite extensions of their centres. For such skew fields it ought to be possible to find some useful notion of transcendence degree over the central subfield k which would give us some idea of the coarse skew subfield structure of such skew fields and ideally would be the maximal length of chains of infinite extensions starting at k and finishing at the skew field. Many of the questions in the following list are based on attempts to set up such a theory.