Finitely generated pathological extensions of difference fields
Finitely generated pathological extensions of difference fields
复制标题
差异场的有限生成病理扩展
DOI：
10.1090/s0002-9947-1962-0133326-0
复制
发表时间：
1962
影响因子：
1.3
通讯作者：
A. Babbitt
中科院分区：
文献类型：
--
作者：
A. Babbitt
3C被称为w的一元扩展，为了简洁起见，我们使用“病态”一词来表示与ff的某些其他扩展不相容或是w的一元扩展的扩展。病态扩展的存在对差分方程的抽象理论和解析理论都有意义。例如，当我们考察与专门化有关的差分代数中的定理时，就会清楚地认识到不相容扩展的基本重要性。事实上，代数几何中关于专门化的标准定理，只有当人们施加排除不相容的条件时，才适用于差分代数。另一方面，一元扩展的存在严重地威胁到为不同领域构造伽罗瓦理论的任务的复杂化。因此，病理扩展对差分代数的意义无疑是巨大的，但我们对这些扩展的认识是相当有限的。Cohn[5]已经证明，一个差分场只有在允许0阶的病态扩展时才允许有限生成的病态扩展，即在地面场上的代数扩展。因此，代数闭域不允许有有限生成的病态扩展。这些是迄今为止获得的关于有限生成的病理扩展的唯一一般结果。鉴于Cohn的结果，很自然地要问是否存在有限生成的零阶病态扩展反过来意味着在地面场上存在有限度病态扩展。我们在本文中的主要目的是调查这一假设；我们的主要结果是这个假设是正确的。在吗?证明了阶的正规扩展的一个分解定理
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