Computing the torsion of the $p$-ramified module

We fix a prime number $p$ and $\text{K}$ a number field, we denote by $M$ the maximal abelian $p$-extension of $\text{Ko}$ unramified outside $p$. The aim of this paper is to study the ${\mathbb{Z}}_p$-module $\text{gal}(M\mathrm{/}\text{Ko})$ and to give a method to effectively compute its structure as a ${\mathbb{Z}}_p$-module. Then we give numerical results, for real quadratic fields, together with interpretations via Cohen-Lenstra’s heuristics.