Questions about algebraic properties of real numbers

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Abstract :

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.
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.
Within constructive analysis, it should be interesting to drop dependent choice (see Richman). 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''.