Abstract: Let $p>2$ be a prime number and $K$ be a number field. Let $S$ be a finite set of primes of $K$ and let $K_S$ be the maximal pro-$p$ extension of $K$ unramified outside $S$; put $G_S=$ Gal $({K_S}{K})$. If $S$ contains the primes above $p$, we know that $cd(G_S)$ less than or equal to $2$, but what is going on if this is not the case?

Thanks to a criteria of Labute, Mináč and Schmidt, we can exhibit mild pro-$p$ groups $G_S$ (and then of cohomological dimension $2$). In this talk I will explain the question of the propagation of the mildness property along some quadratic extensions of $\mathbb{Q}$. In particular, I will give some statistics and some theoretical results.