Note on 2-rational fields

We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field.
Comment: This Note is motivated by the paper ``Galois 2-extensions unramified outside 2'' of J. Jossey. We bring into focus some classical technics on abelian $\ell$-ramification which considerably simplify proofs in such subjects; for instance, the main Theorem 2, due to J-F. Jaulent, generalizes the purpose of Jossey's paper