Groups definable in separably closed fields

Authors:
E. Bouscaren and F. Delon

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

Published electronically:
October 24, 2001

MathSciNet review:
1867366

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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 is definably isomorphic to the group of -rational points of an algebraic group defined over .

**[BoDe]**E. Bouscaren and F. Delon,*Minimal groups in separably closed fields*, to appear in the Journal of Symbolic Logic.**[De 88]**F. Delon,*Idéaux et types sur les corps séparablement clos*, Supplément au Bulletin de la SMF, Mémoire 33, Tome 116 (1988). MR**90m:03067****[De 98]**F. Delon,*Separably closed fields*, in Model Theory and Algebraic Geometry, E. Bouscaren (Ed.), Lecture Notes in Mathematics 1696, Springer, 1998. MR**2000a:12011****[Hu 87]**J.E. Humphreys,*Linear Algebraic Groups*, Graduate Texts in Mathematics, Springer, 1987. MR**53:633****[Hr 96]**E. Hrushovski,*The Mordell-Lang conjecture for function fields*, Journal AMS 9 (1996), 667-690. MR**97h:11154****[La 58]**S. Lang,*Introduction to algebraic geometry*, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers, New York, 1958. MR**20:7021****[Me 94]**M. Messmer,*Groups and fields interpretable in separably closed fields*, TAMS 344 (1994), 361-377. MR**95c:03086****[Me 96]**M. Messmer,*Some model theory of separably closed fields*, in Model Theory of Fields, Lecture Notes in Logic 5, Springer, 1996. MR**98m:03075****[Pi 96]**A. Pillay,*Geometrical Stability Theory*, Oxford University Press, 1996. MR**98a:03049****[Pi 98]**A. Pillay,*Model theory of algebraically closed fields*, in Model Theory and Algebraic Geometry, E. Bouscaren (Ed.), Lecture Notes in Mathematics 1696, Springer, 1998. MR**2000f:12008****[Po 87]**B. Poizat,*Groupes Stables*, Nur al-mantiq wal ma'rifah, Villeurbanne, France, 1987. MR**89b:03056****[Sp 98]**T.A. Springer,*Linear algebraic groups*, 2nd edition, Birkhäuser, 1998. MR**99h:20075****[Wa 97]**F. Wagner,*Stable Groups*, London Math. Soc. LNS 240, Cambridge University Press, 1997. MR**99g:20010****[We 55]**A. Weil,*On algebraic groups of transformations*, American Journal of Math. 77 (1955), 355-391. MR**17:533f**

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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

DOI:
https://doi.org/10.1090/S0002-9947-01-02886-0

Keywords:
Separably closed fields,
groups

Received by editor(s):
January 10, 1999

Received by editor(s) in revised form:
September 20, 2000

Published electronically:
October 24, 2001

Article copyright:
© Copyright 2001
American Mathematical Society