Differential Galois theory

In mathematics, differential Galois theory studies the Galois groups of differential equations.
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 "logarithmlike" 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 an elementary 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.
References
 Bertrand, D. (1996), "Review of "Lectures on differential Galois theory"", Bulletin of the American Mathematical Society 33 (2), ISSN 00029904, http://www.ams.org/bull/19963302/S0273097996006520/S0273097996006520.pdf
 Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series, 7, Providence, R.I.: American Mathematical Society, ISBN 9780821870044, MR1301076, http://books.google.com/books?id=cJ9vByhPqQ8C
 Magid, Andy R. (1999), "Differential Galois theory", Notices of the American Mathematical Society 46 (9): 1041–1049, ISSN 00029920, MR1710665, http://www.ams.org/notices/199909/feamagid.pdf
 van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Berlin, New York: SpringerVerlag, ISBN 9783540442288, MR1960772, http://www4.ncsu.edu/~singer/ms_papers.html
See also
 Liouville's theorem (differential algebra)
 Risch algorithm
Categories: Field theory
 Differential algebra
 Differential equations
 Algebraic groups
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