On the cohomological dimension of some pro-p-extensions above the cyclotomic Z_p-extension of a number field

International audienceLet $\KST$ be the maximal pro-$p$-extension of the cyclotomic $\Z_p$-extension $\K^{cyc}$ of a number field $\K$, unramified outside the places above $S$ and totally split at the places above $T.$ Let $\GST=\Gal(\KST/\K).$ In this work we adapt the methods developed by Schmidt in \cite{schmidt2} in order to show that the group $\GST=\Gal(\KST/\K)$ is of cohomological dimension $2$ provided the finite set $S$ is well chosen. This group $\GST$ is in fact \textit{mild} in the sense of Labute \cite{labute}. We compute its Euler characteristic, by studying the Galois cohomology groups $H^i(\GST,\fq_p)$, $i=1,2.$ Finally, we provide new situations where the group $\GST$ is a free pro-$p$-group