p-Rational Fields and the Structure of Some Modules

Assume that the field $K$ is $p$-rational. We study the freeness of the $\mathrm{\Lambda }(G_{\mathrm{\infty },S})$-module $$\mathcal{X}$=$\mathcal{H}$^{ab}=$\mathrm{G}\mathrm{a}\mathrm{l}$(K_{S\cup S_p}/K_{\infty,S})^{ab}$. For numerical evidence to our result we consider the case of fields of the form $\mathbb{Q}(\sqrt{pq},\sqrt{-d})$.