Fields with few types

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. In model theory, much attention has been drawn on algebraic structures which lie on the classifiable part of the dividing line provided by Shelah's stability theory, following the criterion: the models of some fixed uncountable cardinality of a theory T should not be classifiable if there are too many of them, such as if T is unstable, or stable non-superstable. Concerning theories with few countable models, less is known, even for N0-categorical groups (who have only one countable model up to isomorphism). All the less for the so called small structures who include all possible theories having fewer than continuum many countable models. Known results about small structures concern type-definable equivalence relations [15, 8,9], abelian groups [21], fields [24] and classes of modules [20]. Some have been studied under additional model-theoretic hypothesis: small stable groups [21] and more generally *R-groups [23, chapter 5], No-categorical supersimple groups [3], small one-based structures [14] and small structures with finite coding [5]. Weakly small structures are introduced by Belegradek in [24] to provide a broad framework for both small and minimal structures: they include omega-stable structures but also Ko-categorical, minimal, and af-minimal ones recently introduced by Poizat in [19]. In [24], Wagner shows that infinite small fields are algebraically closed, making the first successful use of the Cantor-Bendixson rank (the very weak analogue of Morley rank) in an algebraic context. He asks whether infinite weakly small fields are algebraically closed [24, Problem 12.5]. Following ideas of [24], an exploration of weakly small groups begins in [13] where it is noticed that a weakly small group G inherits locally several properties that omega-stable groups share globally. For instance G satisfies local descending chain conditions. Every definable subset of Received April 1,2011. 2010 Mathematics Subject Classification. 03C45, 03C60.