Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Groups and fields interpretable in separably closed fields


Author: Margit Messmer
Journal: Trans. Amer. Math. Soc. 344 (1994), 361-377
MSC: Primary 03C60; Secondary 12L12, 20G99
DOI: https://doi.org/10.1090/S0002-9947-1994-1231337-6
MathSciNet review: 1231337
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that any infinite group interpretable in a separably closed field F of finite Eršov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F.


References [Enhancements On Off] (What's this?)

  • [1] A. Borel, Linear algebraic groups, 2nd ed., Springer-Verlag, Berlin and New York, 1991. MR 1102012 (92d:20001)
  • [2] E. Bouscaren, Model-theoretic version of Weil's theorem on pregroups, The Model Theory of Groups (A. Nesin and A. Pillay, eds.), Univ. of Notre Dame Press, 1989, pp. 177-185. MR 985345
  • [3] Z. Chatzidakis, G. Cherlin, S. Shelah, G. Srour, and C. Wood, Orthogonality in separably closed fields, Classification Theory (J. Baldwin, ed.), Springer, New York and Berlin, 1985, pp. 72-88. MR 1033023 (91f:03074)
  • [4] G. Cherlin and S. Shelah, Superstable fields and groups, Ann. Math. Logic 18 (1980), 227-270. MR 585519 (82c:03045)
  • [5] F. Delon, Ideaux et types sur les corps séparablement clos, Suppl. Bull. Soc. Math. France, Mém. 33, Tome 116 (1988). MR 986208 (90m:03067)
  • [6] J.-L. Duret, Les corps pseudo-finis ont la propriété d'indépendance, C. R. Acad. Sci. Paris Sér. I Math. 290 (1980), 981-983. MR 584282 (81m:03043)
  • [7] Ju. L. Eršov, Fields with a solvable theory, Dokl. Akad. Nauk SSSR 174 (1967), 19-20; English transl., Soviet Math. Dokl. 8 (1967), 575-576. MR 0214575 (35:5424)
  • [8] J. E. Humphreys, Linear algebraic groups, Springer-Verlag, Berlin and New York, 1975. MR 0396773 (53:633)
  • [9] N. Jacobson, Lectures in abstract algebra. III, Van Nostrand, Princeton, NJ, 1964. MR 0172871 (30:3087)
  • [10] A. Macintyre, On $ {\omega _1}$-categorical theories of fields, Fund. Math. 71 (1971), 1-25. MR 0290954 (45:48)
  • [11] A. Pillay, An introduction to stability theory, Clarendon Press, Oxford, 1983. MR 719195 (85i:03104)
  • [12] B. Poizat, Groupes stables, Nur alMantiq walMa'arifah, Villeurbanne, 1987. MR 902156 (89b:03056)
  • [13] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443. MR 0082183 (18:514a)
  • [14] G. Srour, The independence relation in separably closed fields, J. Symbolic Logic 51 (1986), 715-725. MR 853851 (87m:03047)
  • [15] L. P. D. van den Dries, Definable groups in characteristic 0 are algebraic groups, Abstracts Amer. Math. Soc. 3 (1982), 142.
  • [16] -, Weil's group chunk theorem: a topological setting, Illinois J. Math. 34 (1990), 127-139. MR 1031890 (91k:22003)
  • [17] A. Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955), 203-271. MR 0074083 (17:533e)
  • [18] C. Wood, Notes on the stability of separably closed fields, J. Symbolic Logic 44 (1979), 412-416. MR 540671 (81m:03042)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C60, 12L12, 20G99

Retrieve articles in all journals with MSC: 03C60, 12L12, 20G99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1231337-6
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society