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Minimal groups in separably closed fields

Published online by Cambridge University Press:  12 March 2014

E. Bouscaren  and
F. Delon
E. Bouscaren
Affiliation:
Université Paris, 7 - CNRS, UFR de Mathématiques - Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France, E-mail: elibou@logique.jussieu.fr
F. Delon
Affiliation:
Université Paris, 7 - CNRS, UFR de Mathématiques - Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France, E-mail: delon@logique.jussieu.fr
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Abstract

We give a complete description of minimal groups infinitely definable in separably closed fields of finite degree of imperfection. In particular we answer positively the question of the existence of such a group with infinite transcendence degree (i.e., a minimal group with non thin generic).

Keywords

Separably closed fieldsgroupsminimal types
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 239 - 259
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[Bo98]Bousearen, E., Proof of the Mordell-Lang conjecture for function fields, Model theory and algebraic geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, 1998.CrossRefGoogle Scholar
[BoDe]Bouscaren, E. and Delon, F., Groups definable in separably closed fields, to appear in the Transactions of the American Mathematical Society.Google Scholar
[Bu96]Buechler, S., Essential stability theory, Perspectives in Mathematical Logic, Springer-Verlag, 1996.CrossRefGoogle Scholar
[ChWo]Chatzidakis, Z. and Wood, C., Minimal types in separably closed fields, to appear in this Journal.Google Scholar
[De88]Delon, F., Idéaux et types sur les corps séparablement clos, Mémoires de la Société Mathématique de France, 33, 1988.Google Scholar
[De98]Delon, F., Separably closed fields, Model theory and algebraic geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, 1998.CrossRefGoogle Scholar
[Hr87]Hrushovski, E., Locally modular regular types, Classification theory (Baldwin, J., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, 1987.CrossRefGoogle Scholar
[Hr96]Hrushovski, E., The Mordell-Lang conjecture for function fields, Journal of the American Mathematical Society, vol. 9 (1996), pp. 667–690.CrossRefGoogle Scholar
[HrSo]Hrushovski, E. and Sokolovic, Z., Minimal subsets of differentially closedfields, to appear in the Transactions of the American Mathematical Society.Google Scholar
[HrZi96]Hrushovski, E. and Zilber, B., Zariski geometries, Journal of the American Mathematical Society, vol. 9 (1996), pp. 1–56.CrossRefGoogle Scholar
[Hu87]Humphreys, J., Linear algebraic groups, Graduate texts in mathematics, Springer-Verlag, 1987.Google Scholar
[La58]Lang, S., Introduction to algebraic geometry, 1958, Interscience.Google Scholar
[La59]Lang, S., Abelian varieties, 1959, Interscience.Google Scholar
[Ma96]Marker, D., Model theory of differential fields, Model theory of fields, Marker, D., Messmer, M., and Pillay, A., Lecture Notes in Logic, vol. 5, Springer, 1996.CrossRefGoogle Scholar
[Ma98]Marker, D., Zariski geometries, Model theory and algebraic geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, 1998.CrossRefGoogle Scholar
[Me94]Messmer, M., Groups and fields interprétable in separably closed fields, Transactions of the American Mathematical Society, vol. 344 (1994), pp. 361–377.CrossRefGoogle Scholar
[Me96]Messmer, M., Some model theory of separably closed fields, Model theory of fields, Marker, D., Messmer, M., and Pillay, A., Lecture Notes in Logic, vol. 5, Springer, 1996.CrossRefGoogle Scholar
[Pi96a]Pillay, A., Differential algebraic groups and the number of countable differentially closed fields, Model theory of fields, Marker, D., Messmer, M., and Pillay, A., Lecture Notes in Logic, vol. 5, Springer, 1996.CrossRefGoogle Scholar
[Pi96b]Pillay, A., Geometrical stability theory, Oxford University Press, 1996.CrossRefGoogle Scholar
[Pi98]Pillay, A., Model theory of algebraically closed fields, Model theory and algebraic geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, 1998.Google Scholar
[Po87]Poizat, B., Groupes stables, Nur al-mantiq wal ma'rifah, Villeurbanne, France, 1987.Google Scholar
[Sp94]Springer, T. A., Linear algebraic groups, Algebraic geometry IV (Parshin, A.N. and Shafarevich, I.R., editors), Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1994.Google Scholar
[Ti67]Tits, J., Lectures on algebraic groups, Yale University, Fall term 1966–1967.Google Scholar
[Wa97]Wagner, F., Stable groups, London Mathematical Society Lecture Note Series, vol. 240, Cambridge University Press, 1997.Google Scholar
[We48]Weil, A., Courbes algébriques et variétés abéliennes, Hermann, Paris, 1948.Google Scholar
[Wo98]Wood, C., Differentially closed fields, Model theory and algebraic geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, 1998.CrossRefGoogle Scholar