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.

See also

• Picard–Vessiot theory

References

• Hubbard, John H.; Lundell, Benjamin E. (2011). "A First Look at Differential Algebra" (PDF). The American Mathematical Monthly. 118 (3): 245–261. doi:10.4169/amer.math.monthly.118.03.245. JSTOR 10.4169/amer.math.monthly.118.03.245. S2CID 1567399.
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• 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
• Juan J. Morales Ruiz : Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser, 1999, ISBN 978-3764360788 .