Differential Galois theory: Difference between revisions

Content deleted Content added

Inline

Revision as of 07:55, 30 December 2016

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 not addressed by differential Galois theory, but is solved by Liouville's theorem and the Risch algorithm.

References

Bertrand, D. (1996), "Review of "Lectures on differential Galois theory"" (PDF), Bulletin of the American Mathematical Society, 33 (2), doi:10.1090/s0273-0979-96-00652-0, ISSN 0002-9904
Beukers, Frits (1992), "8. Differential Galois theory", in Waldschmidt, Michel; Moussa, Pierre; Luck, Jean-Marc; Itzykson, Claude (eds.), From number theory to physics. Lectures of a meeting on number theory and physics held at the Centre de Physique, Les Houches (France), March 7–16, 1989, Berlin: Springer-Verlag, pp. 413–439, ISBN 3-540-53342-7, Zbl 0813.12001
Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series, vol. 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-7004-4, MR 1301076
Magid, Andy R. (1999), "Differential Galois theory" (PDF), Notices of the American Mathematical Society, 46 (9): 1041–1049, ISSN 0002-9920, MR 1710665
van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44228-8, MR 1960772

@@ Line 2: / Line 2: @@
 ==Overview==
-Whereas algebraic [[Galois theory]] studies extensions of [[field (mathematics)|algebraic fields]], differential Galois theory studies extensions of [[differential field]]s, i.e. fields that are equipped with a [[derivation (abstract algebra)|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 [[integral]]s of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of [[polynomial equation]]s by [[Nth root|radicals]] in algebraic Galois theory, and is solved by [[Picard–Vessiot theory]].
+Whereas algebraic [[Galois theory]] studies extensions of [[field (mathematics)|algebraic fields]], differential Galois theory studies extensions of [[differential field]]s, i.e. fields that are equipped with a [[derivation (abstract algebra)|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 [[integral]]s of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of [[polynomial equation]]s by [[Nth root|radicals]] in algebraic Galois theory, and is not addressed by differential Galois theory, but is solved by [[Liouville's theorem (differential algebra)|Liouville's theorem]] and the [[Risch algorithm]].
 ==See also==
 *[[Liouville's theorem (differential algebra)]]
+*[[Picard-Vessiot theory]]
 *[[Risch algorithm]]