Totally real closures of fields of formally real fields

This article is concerned with arithmetic properties of totally real closur es of formally real fields. We generalize previous results of Fried, Volkle in 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 descr iption of the Brauer group of (K) over tilde(tr): it is always an elementar y abelian 2-group. Finally, using a result of Glass and Ribenboim, we show that an automorphism of the group Aut((K) over tilde(tr)(+)*, less than or equal to 1..), where K is a formally real number field, is necessarily the identity.