Abstract
Separably closed fields are stable. When they are not algebraically closed, they are rather complicated from a model theoretic point of view: they are not super-stable, they admit no non trivial continuous rank and they have the dimensional order property. But they have a fairly good theory of types and independence, and interesting minimal types. Hrushovski used separably closed fields in his proof of the Mordell-Lang Conjecture for function fields in positive characteristic in the same way he used differentially closed fields in characteristic zero ([Hr 96], see [Bous] in this volume). In particular he proved that a certain class of minimal types, which he called thin, are Zariski geometries in the sense of [Mar] section 5. He then applied to these types the strong trichotomy theorem valid in Zariski geometries.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. Bourbaki, XI, Algèbre, chapitre 5, Corps commutatifs, Hermann, Paris 1959.
E. Bouscaren, Proof of the Mordell-Lang conjecture for function fields, this volume.
C. Chang and J. Keisler, Model Theory, North-Holland, Amsterdam 1973.
Z. Chatzidakis, G. Cherlin, S. Shelah, G. Srour and C. Wood, Orthogonality of types in separably closed fields, in Classification Theory (Proceedings of Chicago, 1985), LNM 1292, Springer, 1987.
Z. Chatzidakis and C. Wood, manuscript, 1996.
F. Delon, Idéaux et types sur les corps séparablement clos, Supplément au Bulletin de la SMF, Mémoire 33, Tome 116 (1988).
Y. Ershov, Fields with a solvable theory, Sov. Math. Dokl. 8 (1967), 575–576.
E. Hrushovski, Unidimensional theories are superstable, APAL 50 (1990), 117–138.
E. Hrushovski, The Mordell-Lang conjecture for function fields, J.AMS 9 (1996), 667–690.
S. Lang, Introduction to algebraic Geometry, Interscience Tracts in pure and applied Mathematics, Interscience Publishers, New York 1958.
65] S. Lang, Algebra, Addison-Wesley, 1965.
D. Marker, Zariski geometries, this volume.
M. Messmer, Groups and fields interpretable in separably closed fields, TAMS 344 (1994), 361–377.
M. Messmer, Some model theory of separably closed fields, in Model Theory of Fields, Lecture Notes in Logic 5, Springer, 1996.
A. Pillay, Model theory of algebraically closed fields, this volume.
A. Weil, Courbes algébriques et variétés abéliennes, Hermann, Paris, 1948.
C. Wood, Notes on the stability of separably closed fields, JSL 44 (1979), 412–416.
M. Ziegler, Introduction to stability theory and Morley rank, this volume.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Delon, F. (1998). Separably closed fields. In: Bouscaren, E. (eds) Model Theory and Algebraic Geometry. Lecture Notes in Mathematics, vol 1696. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68521-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-68521-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64863-5
Online ISBN: 978-3-540-68521-0
eBook Packages: Springer Book Archive