On extensions of number fields with given quadratic algebras and cohomology

TL;DR

This paper develops a criterion for analyzing the cohomology of finitely presented pro-p groups, enabling the inference of properties of Galois groups over p-rational fields with specific ramification, advancing understanding of their algebraic structure.

Contribution

It introduces a new criterion for presenting pro-p groups that facilitates cohomology computations and the analysis of Galois groups with prescribed ramification.

Findings

Computed cohomology groups for certain pro-p groups

Inferred properties of Galois groups over p-rational fields

Extended understanding of group quotients with higher cohomological dimension

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.