Isotypic components of pro-p-number body extensions and p-rationality

Let p be a prime number, and let K/k be a finite Galois extension of number fields with Galois group Delta of order coprime to p. Let S be a finite set of non-Archimedean places of k including the set S-p of p-adic places, and let KS be the maximal pro-p extension of K unramified outside S. Let G := G(S)/H be a quotient of GS := Gal(K-S/K) on which Delta acts trivially. Put chi := H[H, H]. In this paper, we study the phi-component chi(phi) of chi for all Qp-irreductible characters phi of Delta, and, in particular, by assuming the Leopoldt conjecture, we show that for all non-trivial characters phi, the Z(p)[G]-module chi(phi) is free if and only if the phi-component of the Z(p)-torsion of G(S)/[G(S),G(S)] is trivial. We also make a numerical study of the freeness of X-phi in cyclic extensions K/Q of degree 3 and 4 (by using families of polynomials given by Balady, Lecacheux, and more recently by Balady and Washington), but also in degree 6 dihedral extension over Q: the results we get support a recent conjecture of Gras.