A recursive description of pro-p Galois groups

In this note we extend the results of our earlier work ``Kaplansky's radical and a recursive description of pro-2 Galois groups'' (Rel. Pesq. 23/01) to arbitrary prime numbers $p$. Although we succeed in proving the same results, the methods used in the proofs are more conceptual. To be precise, let $\G{F}$ be the Galois group of the maximal Galois $p$-extension of a field$F$ of characteristic $\ne p$. Denote by $R(F)$ the radical of the skew-symmetric bilinear pairing which associates to each pair $a$, $b$ of non-zero elements of $F$ the class of the cyclic algebra $(a,b)_{F}$ in the Brauer group of $F$. We deduce from a condition connecting $R(F)$ with valuation rings of $F$ and also orderings of $F$ when $p = 2$, that $\G{F}$ can be obtained from some suitable closed subgroups using free pro-$2$ products and semi-direct group extension operations a finite number of times.