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

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