Differential Galois Theory

Zusammenfassung

The differential Galois theory for linear differential equations is the Picard- Vessiot Theory. In this theory there is a very nice concept of “integrability” i.e., solutions in closed form: an equation is integrable if the general solution is obtained by a combination of algebraic functions (over the coefficient field), exponentiation of quadratures and quadratures. Furthermore, all information about the integrability of the equation is coded in the identity component of the Galois group: the equation is integrable if, and only if, the identity component of its Galois group is solvable. It is a powerful theory in the sense that, in some favorable cases (for instance, for equations of order 2), it is possible to construct algorithms to determine whether a given linear differential equation is integrable or not.