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

Abstract

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 mild, hence of cohomological dimension 2.