Differential Galois Theory: exp(-x^2) has no elementary antiderivative

[kowalski]

A lot of apparently innocent elementary functions, like exp(-x^2) or (sin x)/x, have not antiderivatives in terms of elementary functions. I've read that "Differential Galois theory" explains this, and gives an algorithmic method to know if a given elementary function has or has not elementary antiderivative.
Please, can you explain to me the fundamental, core ideas of this theory?. Some practical, as elementary as possible references? Examples of its use?. Thank you, kowalski.

 

[Cygni]

Hey Kowalski,

i found some decent papers about galois theory. You may find them on this website http://www4.ncsu.edu/~singer/ms_papers.html

 

[kowalski]

Hola Cygni,
thanks for the references; if I learn how to apply effectively this theory (which is my goal) I will post a resume. Thanx again.