Simple axiomatizations of Rolle fields
An ordered field \(K\) is said to have Rolle's property if for every polynomial \(P\in K[X]\) and for every \(a,b\in K\), \(a<b\), such that \(P(a)=P(b)=0\) there exists \(c\in K\), \(a<c<b\), for which \(P'(c)=0\). If \(K\) has Rolle's property for some ordering it has Rolle's property for every ordering. Such ordered field is called a Rolle field [\textit{R. Brown}, \textit{T. C. Craven} and \textit{M. J. Pelling}, Ill. J. Math. 30, 66--78 (1986; Zbl 0559.12013)]. The author formulates a first order axiomatization for the theory of Rolle fields, similar to the Artin-Schreier axiomatization. Rolle fields with a finite number of orderings are investigated. They are characterized as Pythagorean fields at level 4 which do not admit any algebraic extension of odd degree. An axiomatization for Rolle fields having exactly \(2^n\) orderings is presented. It is shown that for every Rolle field \(K\) with exactly \(2^n\) orderings there exist such orders \(P_0,\ldots, P_n\) that \(K=R_0\cap\cdots\cap R_n\), \(R_i\) being the real closure of the field \(K\) corresponding to the order \(P_i\). This theorem may be considered as a generalization of a result of \textit{E. Becker} [Hereditarily-Pythagorean fields and orderings of higher level, Monogr. Mat. 29 (1978; Zbl 0509.12020)].
scientific article; zbMATH DE number 4177219