p-Rational Fields and the Structure of Some Modules

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