  In this memoir, we seek to construct a dynamical theory as complete as
possible to describe the algebraic properties of the field of real numbers in
constructive mathematics without axiom of dependent choice. We propose a theory
which turns out to be very close to the theory of real closed local rings in
classical mathematics. The theory of real closed rings is presented here in
constructive form as a natural purely equational theory, which uses virtual
root functions introduced in previous work. This work is also a first step
through an essential goal for the future, which is to obtain a constructive
version of o-minimal structures.
