Geodesic
Splitting fields and general differential Galois theory
Sbornik. Mathematics, Tome 201 (2010) no. 9, pp. 1323-1353 Cet article a éte moissonné depuis la source Math-Net.Ru

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An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions. Bibliography: 14 titles.
Keywords: tensor products, constructed fields, differential closure, splitting field, differential Galois group. 
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D. V. Trushin. Splitting fields and general differential Galois theory. Sbornik. Mathematics, Tome 201 (2010) no. 9, pp. 1323-1353. http://geodesic.mathdoc.fr/item/SM_2010_201_9_a3/

[1] E. R. Kolchin, “Constrained extensions of differential fields”, Advances in Math., 12:2 (1974), 141–170 | DOI | MR | Zbl

[2] S. Shelah, “Uniqueness and characterization of prime models over sets for totally transcendental first-order theories”, J. Symbolic Logic, 37:1 (1972), 107–113 | DOI | MR | Zbl

[3] B. Poizat, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985 | MR | Zbl

[4] M. van der Put, M. F. Singer, Galois theory of linear differential equations, Grundlehren Math. Wiss., 328, Springer-Verlag, Berlin, 2003 | MR | Zbl

[5] Ph. J. Cassidy, M. F. Singer, “Galois theory of parameterized differential equations and linear differential algebraic groups”, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, 2007, 113–157 | MR | Zbl

[6] A. Pillay, “Differential Galois theory. I”, Illinois J. Math., 42:4 (1998), 678–699 | MR | Zbl

[7] A. Pillay, “Differential Galois theory. II”, Ann. Pure Appl. Logic, 88:2–3 (1997), 181–191 | DOI | MR | Zbl

[8] D. Marker, A. Pillay, “Differential Galois theory. III: Some inverse problems”, Illinois J. Math., 41:3 (1997), 453–461 | MR | Zbl

[9] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA–London–Don Mills, ON, 1969 | MR | MR | Zbl | Zbl

[10] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York–London, 1973 | MR | Zbl

[11] J. J. Kovacic, “The differential Galois theory of strongly normal extensions”, Trans. Amer. Math. Soc., 355:11 (2003), 4475–4522 | DOI | MR | Zbl

[12] Th. Scanlon, “Model theory and differential algebra”, Differential algebra and related topics (Newark, NJ, 2000), World Sci. Publ., Singapore–River Edge, NJ, 2002, 125–150 | MR | Zbl

[13] T. McGrail, “The model theory of differential fields with finitely many commuting derivations”, J. Symbolic Logic, 65:2 (2000), 885–913 | DOI | MR | Zbl

[14] M. Rosenlicht, “The nonminimality of the differential closure”, Pacific J. Math., 52:2 (1974), 529–537 | MR | Zbl