algebra See algebra. abstraction An abstraction of a concept C is a concept C ′ that includes all instances of C and that is constructed by taking as axioms certain assertions that are true of all instances of C. C may already be defined mathematically, in which case the abstraction is typically a legitimate generalization of C. In other cases, C may be a familiar concept or property that has not been given a mathematical definition. In that case, the mathematical definition may allow instances of the abstract version of C that were not originally thought of as being part of C.ion An abstraction of a concept C is a concept C ′ that includes all instances of C and that is constructed by taking as axioms certain assertions that are true of all instances of C. C may already be defined mathematically, in which case the abstraction is typically a legitimate generalization of C. In other cases, C may be a familiar concept or property that has not been given a mathematical definition. In that case, the mathematical definition may allow instances of the abstract version of C that were not originally thought of as being part of C. Example 1 The concept of “group” is historically an abstraction of the concept of the set of all symmetries of an object. The group axioms are all true assertions about symmetries when the binary operation is taken to be composition of symmetries. Example 2 The -δ definition of continuous function is historically an abstraction of the intuitive idea that mathematicians had about functions that there was no “break” in the output. This abstraction became the standard definition of “continuous”, but allowed functions to be called continuous that were not contemplated before the definition was introduced. Other examples are given under model and in Remark 2 under free variable. See also the discussions under definition, generalization and representation. Citations: (31), (270). References: [Dreyfus, 1992], [Thompson, 1985].

A Handbook of Mathematical Discourse

A Galois theory for differential fields

We study groups defined by algebraic difference equations. These groups occur as the Galois groups of linear differential and difference equations depending on discrete parameters. Among the main results are three finiteness theorems, the introduction of numerical invariants such as the limit degree, a dimension theorem for difference algebraic groups, an analog of Chevalley's theorem on representations and an analog of the Jordan-H\"{o}lder theorem.

Affine difference algebraic groups

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulae associated with finite Galois covers of difference schemes.

Twisted Galois stratification

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

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Finitely generated pathological extensions of difference fields

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Finitely generated pathological extensions of difference fields