Differential Galois theory

In mathematics, differential Galois theory studies the Galois groups of differential equations.

Overview

Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. The problem of finding which integrals of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals in algebraic Galois theory, and is solved by Picard–Vessiot theory.

Definitions

For any differential field F, there is a subfield

Con(F) = {f in F | Df = 0},

called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that

Dt = Ds/s for some s in F.

This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension which satisfies

Dt = tDs.

With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called a Liouvillian differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.

See also

• Liouville's theorem (differential algebra)
• Risch algorithm