Questions about algebraic properties of real numbers

This paper is a survey of natural questions (with few answers) arising when one wants to study algebraic properties of real numbers, i.e., properties of real numbers w.r.t. {+, −, ×, >, ≥} in a constructive setting. Introduction This paper is a survey of natural questions (with few answers) arising when one wants to study algebraic properties of real numbers, i.e., properties of real numbers w.r.t. {+, −, ×, >, ≥} in a constructive setting (see [1, 12]). Why studying constructive real algebra? A first reason is that constructive real algebra is not well understood! Constructive analysis is much more developped. From a constructive point of view, real algebra is far away from the theory of discrete real closed fields (which was settled by Artin in order to understand real algebra in the framework of classical logic). Most algorithms for discrete real closed fields fail for Dedekind real numbers, because we have no sign test for real numbers. Another reason is that within constructive analysis, it should be interesting to drop dependent choice (see [13]). A study of real agebra without dependent choice could help. Last but not least, understanding constructive real algebra should be a first important step towards a constructive version of O-minimal structures. Real algebra can be seen instead as the simplest O-minimal structure. Indeed classical O-minimal structures give effectiveness results inside classical mathematics, but they are not completely effective, because the sign test on real numbers is needed for the corresponding “algorithms”. Finally let us mention that we try also to propose a theory developped without using logical absurdity. Fred Richman shows that constructive mathematics are more elegant without dependent choice. The author thinks also that they are more elegant without logical absurdity. Let us remark for example that “Ex falso quodlibet” is easily replaced by “from 1 = 0 you can prove any positive fact in a commutative ring”. Another example: in constructive commutative algebra many theorems become simpler if we allow the trivial ring to be a local ring and a discrete field. E.g., a quotient of a local ring is a local ring, even if we don’t know if the quotient is trivial or not. Acknowlegment: I’m grateful to a referee for his (her) very useful comments and suggestions. ∗ Equipe de Mathématiques, UMR CNRS 6623, UFR des Sciences and Techniques, Université de FrancheComté, 25030 BESANCON cedex, FRANCE, email: [email protected]