On extensions of number fields with given quadratic algebras and cohomology

Abstract

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, from (non-free) mild groups. We interpret these groups as Galois groups over $p$-rational fields with prescribed ramification and splitting.

18 pages, Final version, To appear in Manuscripta Mathematica