p-Rational Fields and the Structure of Some Modules

Abstract

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