Differential Galois theory: Difference between revisions
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==Overview== |
==Overview== |
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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 solved by [[Picard–Vessiot theory]]. |
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==Definitions== |
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For any differential field ''F'', there is a subfield |
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:Con(''F'') = {''f'' in ''F'' | ''Df'' = 0}, |
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called the [[constant (mathematics)|constants]] of ''F''. Given two differential fields ''F'' and ''G'', ''G'' is called a '''logarithmic extension''' of ''F'' if ''G'' is a [[field extension|simple transcendental extension]] of ''F'' (i.e. ''G'' = ''F''(''t'') for some [[Transcendental element|transcendental]] ''t'') such that |
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:''Dt'' = ''Ds''/''s'' for some ''s'' in ''F''. |
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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 |
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:''Dt'' = ''tDs''. |
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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. |
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==See also== |
==See also== |
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Revision as of 07:50, 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 solved by Picard–Vessiot theory.
See also
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