Differential Fields

Abstract

We develop in this chapter the algebraic machinery in which the integration algorithms can be presented and proved correct. The main idea, which originates from J. F. Ritt [63], is to define the notion of derivation in a pure algebraic setting (i.e. without using the notions of “function”, “limit”, and “tangent line” from analysis) and to study the properties of such formal derivations on arbitrary objects. This way, we can later translate an integration problem to solving an equation in some algebraic structure, which can be done using algebraic algorithms. Since an arbitrary transcendental function can be seen as a univariate rational function over a field with an arbitrary derivation, we first need to study the general properties of derivations over rings and fields. This will allow us to generalize the rational function integration algorithms to large classes of transcendental functions (Chap. 5).