Clôtures totalement réelles des corps de nombres ordonnables

Abstract:

This article is concerned with arithmetic properties of totally real closures of formally real fields. We generalize previous results of Fried, V\"olklein and Pop to show that if an algebraic extension K/ℚ is formally real and hilbertian then the absolute Galois group of the cyclotomic closure of the totally real closure of K is pro-free. In addition, we give a precise description of the Brauer group of : it is always an elementary abelian 2-group. Finally, using a result of Glass and Ribenboim, we show that an automorphism of the group , where K is a formally real number field, is necessarily the identity.