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Fields with few types

Published online by Cambridge University Press:  12 March 2014

Cédric Milliet
Cédric Milliet*
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS, 43 Boulevard DU 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
Université Galatasaray, Faculté de Sciences et de Lettres, Département de Mathématiques, Çiraǧan Caddesi N 36, 34357 Ortaköy Istamboul, Turquie, E-mail: milliet@math.univ-lyonl.fr
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Abstract

According to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.

Keywords

Smallweakly smallfieldArtin-Schreier extensionCantor-Bendixson ranklocal descending chain condition
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 72 - 84
Copyright
Copyright © Association for Symbolic Logic 2013

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