On extensions of number fields with given quadratic algebras and cohomology

We introduce a criterion on the presentation of finitely presented pro-$p$ groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two. We interpret these groups as Galois groups over $p$-rational fields with prescribed ramification and splitting.Comment: 17 pages, better exposition, removing Proposition 1.1 from previous version, more details in Part