Schlanke Körper (slim fields). (English) Zbl 1218.03027

The authors introduce and study the notion of a (very) slim field. The paper contains many varied results and also presents some open questions on the subject. A field \(K\) (with possible extra structure) is slim if the model-theoretic algebraic closure in \(K\) coincides with the relative field-theoretic algebraic closure. \(K\) is very slim if all fields elementary equivalent to \(K\) are slim (e.g., finite fields, algebraically closed fields, perfect PAC fields, …). In Section 2, the authors show that very slimness is equivalent to algebraic dependence, being an abstract independence relation in the sense of Kim and Pillay, and also to the fact that topological independence (w.r.t. the Zariski topology) coincides with algebraic independence (Theorem 2.1). They also give an equivalent formulation of very slimness (Proposition 2.9). In Section 3, a body mass function is introduced and used to define the size of a field. These notions can be seen as “measuring” the “non-slimness” of a field. For example, very slim fields have size XS while a slim field may well have size S, M or L. Section 4 presents basic model-theoretic and algebraic properties of (very) slim fields. For example, the authors prove that very slim fields are perfect and do not have proper infinite definable subfields (Poposition 4.1) and that elementary substructures of slim fields are slim (Lemma 4.11). They also study the behaviour of finite field extensions (e.g., Propositions 4.6 and 4.7, Corollary 4.10) and questions of axiomatisability for slimness (Question 7). Section 5 contains two important algebraic results. Namely, large model-complete fields and henselian valued fields of characteristic 0 are shown to be very slim (Theorems 5.4 and 5.5). The paper finishes with examples of fields that are slim bot not very slim, and fields that are not slim.


03C60 Model-theoretic algebra
12L12 Model theory of fields
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