arXiv subjects

Abdelaziz El Habibi

Publications and source records attributed to Abdelaziz El Habibi.

2 recordsLinked to original sources

arXiv2022

Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is unramified outside $S$, to be \textit{mild}, hence of cohomological dimension $2$.

math.NT↗

arXiv2018

p-Rational Fields and the Structure of Some Modules

Assume that the field $K$ is $p$-rational. We study the freeness of the $Λ(G_{\infty,S})$-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 $\mathbb{Q}(\sqrt{pq},\sqrt{-d})$.

math.NT↗