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
MathSciNet review: 1231337
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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.


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  • [1] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
  • [2] Elisabeth Bouscaren, Model theoretic versions of Weil’s theorem on pregroups, The model theory of groups (Notre Dame, IN, 1985–1987) Notre Dame Math. Lectures, vol. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 177–185. MR 985345
  • [3] Z. Chatzidakis, G. Cherlin, S. Shelah, G. Srour, and C. Wood, Orthogonality of types in separably closed fields, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 72–88. MR 1033023, 10.1007/BFb0082232
  • [4] G. Cherlin and S. Shelah, Superstable fields and groups, Ann. Math. Logic 18 (1980), no. 3, 227–270. MR 585519, 10.1016/0003-4843(80)90006-6
  • [5] 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
  • [6] Jean-Louis Duret, Les corps pseudo-finis ont la propriété d’indépendance, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 21, A981–A983 (French, with English summary). MR 584282
  • [7] Ju. L. Eršov, Fields with a solvable theory, Dokl. Akad. Nauk SSSR 174 (1967), 19–20 (Russian). MR 0214575
  • [8] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
  • [9] Nathan Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, 1964. MR 0172871
  • [10] Angus Macintyre, On 𝜔₁-categorical theories of fields, Fund. Math. 71 (1971), no. 1, 1–25. (errata insert). MR 0290954
  • [11] Anand Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press, Oxford University Press, New York, 1983. MR 719195
  • [12] Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], 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
  • [13] Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443. MR 0082183
  • [14] G. Srour, The independence relation in separably closed fields, J. Symbolic Logic 51 (1986), no. 3, 715–725. MR 853851, 10.2307/2274025
  • [15] L. P. D. van den Dries, Definable groups in characteristic 0 are algebraic groups, Abstracts Amer. Math. Soc. 3 (1982), 142.
  • [16] L. P. D. van den Dries, Weil’s group chunk theorem: a topological setting, Illinois J. Math. 34 (1990), no. 1, 127–139. MR 1031890
  • [17] André Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955), 355–391. MR 0074083
  • [18] Carol Wood, Notes on the stability of separably closed fields, J. Symbolic Logic 44 (1979), no. 3, 412–416. MR 540671, 10.2307/2273133

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