Groups definable in separably closed fields
HTML articles powered by AMS MathViewer
- by E. Bouscaren and F. Delon PDF
- Trans. Amer. Math. Soc. 354 (2002), 945-966 Request permission
Abstract:
We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field $L$ is definably isomorphic to the group of $L$-rational points of an algebraic group defined over $L$.References
- E. Bouscaren and F. Delon, Minimal groups in separably closed fields, to appear in the Journal of Symbolic Logic.
- Françoise Delon, Idéaux et types sur les corps séparablement clos, Mém. Soc. Math. France (N.S.) 33 (1988), 76 (French, with English summary). MR 986208
- Françoise Delon, Separably closed fields, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 143–176. MR 1678543, DOI 10.1007/978-3-540-68521-0_{9}
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
- Serge Lang, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958. MR 0100591
- Margit Messmer, Groups and fields interpretable in separably closed fields, Trans. Amer. Math. Soc. 344 (1994), no. 1, 361–377. MR 1231337, DOI 10.1090/S0002-9947-1994-1231337-6
- D. Marker, M. Messmer, and A. Pillay, Model theory of fields, Lecture Notes in Logic, vol. 5, Springer-Verlag, Berlin, 1996. MR 1477154, DOI 10.1007/978-3-662-22174-7
- Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
- Anand Pillay, Model theory of algebraically closed fields, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 61–84. MR 1678602, DOI 10.1007/978-3-540-68521-0_{4}
- Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
- Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
- Frank O. Wagner, Stable groups, London Mathematical Society Lecture Note Series, vol. 240, Cambridge University Press, Cambridge, 1997. MR 1473226, DOI 10.1017/CBO9780511566080
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Additional Information
- E. Bouscaren
- Affiliation: Université Paris 7- CNRS, UFR de Mathématiques, Case 7012, 2 Place Jussieu, 75251 Paris cedex 05, France
- Email: elibou@logique.jussieu.fr
- F. Delon
- Affiliation: Université Paris 7- CNRS, UFR de Mathématiques, Case 7012, 2 Place Jussieu, 75251 Paris cedex 05, France
- Email: delon@logique.jussieu.fr
- Received by editor(s): January 10, 1999
- Received by editor(s) in revised form: September 20, 2000
- Published electronically: October 24, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 945-966
- MSC (1991): Primary 03C60, 03C45, 12L12
- DOI: https://doi.org/10.1090/S0002-9947-01-02886-0
- MathSciNet review: 1867366