p-Rational Fields and the Structure of Some Modules

TL;DR

This paper investigates the structure and freeness of certain modules associated with p-rational fields, providing theoretical insights and numerical evidence for specific quadratic fields.

Contribution

It introduces new results on the module structure over p-rational fields and offers numerical evidence for quadratic fields of a particular form.

Findings

Freeness of modules depends on p-rationality of the field

Numerical evidence supports theoretical results for specific quadratic fields

Provides conditions under which modules are free

Abstract

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