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 \(\mathbb{Q}(\sqrt{pq},\sqrt{-d})\).