Abstract
Using Becker's results we obtain here a simple first order axiomatization, looking like those by Artin-Schreier and also written in the language of fields, for the theory of Rolle fields (i.e. fields with the Rolle's property for every order). In fields having a finite number of orders, we characterize Rolle fields as those which are pythagorean at level 4 and do not admit any algebraic extension of odd degree.
Then we give an axiomatization for Rolle fields having exactly 2n orders (n≥0); in fact, for n=0 we recover an axiomatization of the theory of real-closed fields and for n=1 we get exactly an axiomatization given for the theory of chain-closed fields by the author in [G1].
Finally we prove that a Rolle field with exactly 2n orders is the intersection of n+1 real closures of the field.
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Gondard-Cozette, D. Axiomatisations simples des theories des corps de rolle. Manuscripta Math 69, 267–274 (1990). https://doi.org/10.1007/BF02567925
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DOI: https://doi.org/10.1007/BF02567925