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Corps et anneaux de Rolle


Author: Françoise Delon
Journal: Proc. Amer. Math. Soc. 97 (1986), 315-319
MSC: Primary 12L05; Secondary 03C60, 12J10
DOI: https://doi.org/10.1090/S0002-9939-1986-0835889-8
MathSciNet review: 835889
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Abstract: Brown, Craven and Pelling have proved that if the polynomials over an ordered field $ K$ satisfy Rolle's theorem, they satisfy it for any ordering on $ K$. We say that such a field is a Rolle field. We prove that this is a first order property in the language of rings, and that the theory of Rolle fields is decidable. Then we give a common generalisation of these fields and the real closed rings defined by Cherlin and Dickmann: The polynomials over an ordered ring $ A$ satisfy Rolle's theorem iff $ A$ is the valuation ring of a henselian valued field with Rolle residue field and $ m$-divisible value group for all odd $ m$.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0835889-8
Keywords: Completeness, decidability, real closed ring, Rolle's theorem
Article copyright: © Copyright 1986 American Mathematical Society