Corps et anneaux de Rolle

Abstract

Brown, Craven and Pelling have proved that if the polynomials over an ordered field K K satisfy Rolle’s theorem, they satisfy it for any ordering on K 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 A satisfy Rolle’s theorem iff A A is the valuation ring of a henselian valued field with Rolle residue field and m m -divisible value group for all odd m m .