Finitely generated pathological extensions of difference fields
Albert E. Babbitt · Transactions of the American Mathematical Society · 1962
3C is termed a monadic extension of W. For the sake of brevity, we use the term pathological to designate an extension which either is incompatible with some other extension of ff or is a monadic extension of W. The existence of pathological extensions has implications for both the abstract and the analytic theory of difference equations [5]. The fundamental importance of incompatible extensions, for example, becomes clear when one examines the body of theorems in difference algebra concerning specializations. Indeed the standard theorems of algebraic geometry regarding specializations go over to difference algebra only if one imposes conditions precluding incompatibility [6]. The existence of monadic extensions on the other hand, threatens seriously to complicate the task of constructing a Galois theory for difference fields. Thus the significance of pathological extensions to difference algebra is unquestionably great, but our knowledge of these extensions is quite limited. Cohn [5] has shown that a difference field admits finitely generated pathological extensions only if it admits stuch extensions of order zero, i.e. algebraic over the ground field. Hence fields which are algebraically closed admit no finitely generated pathological extensions. These are the only general results regarding finitely generated pathological extensions yet obtained. In view of Cohn's results it is natural to inquire if the existence of finitely generated pathological extensions of order zero in turn implies the existence of pathological extensions which are of finite degree over the ground field. Our primary purpose in this paper is to investigate this supposition; our principal result is that this supposition is true. In ?2 we prove a decomposition theorem for normal extensions of order