Editing Differential Galois theory (section)

== Motivation and basic concepts ==
In [[mathematics]], some types of [[elementary function]]s cannot express the [[indefinite integral]]s of other elementary functions. A well-known example is <math>e^{-x^2}</math>, whose indefinite integral is the [[error function]] <math>\operatorname{erf}x</math>, familiar in [[statistics]]. Other examples include the [[sinc function]] <math>\tfrac{\sin x}{x}</math> and <math>x^x</math>.

It's important to note that the concept of elementary functions is merely conventional. If we redefine elementary functions to include the error function, then under this definition, the indefinite integral of <math>e^{-x^2}</math> would be considered an elementary function. However, no matter how many functions are added to the definition of elementary functions, there will always be functions whose indefinite integrals are not elementary.

Using the theory of '''differential Galois theory''' , it is possible to determine which indefinite integrals of elementary functions cannot be expressed as elementary functions. Differential Galois theory is based on the framework of [[Galois theory]]. While algebraic Galois theory studies [[field extension]]s of [[field (mathematics)|field]]s, differential Galois theory studies extensions of '''[[differential field]]s'''—fields with a '''[[derivation (differential algebra)|derivation]]''' ''D''.

Most of differential Galois theory is analogous to algebraic Galois theory. The significant difference in the structure is that the Galois group in differential Galois theory is an [[algebraic group]], whereas in algebraic Galois theory, it is a [[profinite group]] equipped with the [[Krull topology]].