Special quotients of absolute Galois Groups with Applications in Number Theory and Pythagorean fields
Abstract
This document introduces the work of Hamza and his collaborators during his PhD studies. Hamza works on profinite Galois Theory: more precisely, his work focuses on realisation of pro-$p$ Galois groups over some fields with specific properties (filtrations, cohomology ...) for a fixed prime $p$. This thesis gives a particular attention to number fields and Pythagorean fields.
The first chapter studies connections between the work of Brumer on compact modules and the work of Lazard on filtrations. Using results of Koch and Shafarevich, the previous connections are applied to the theory of pro-$p$ groups and fields extensions. This chapter sets the background to study the rest of the work of Hamza and his collaborators.
The second, third and fourth chapters are papers written by Hamza and his collaborators, in which they investigate extensions of number fields with restricted ramification and non trivial cohomology. The fifth chapter is a work in preparation between Hamza, Maire, Min{\' a}{\v c} and Tân where they introduce a class of pro-$2$ groups that they call $\Delta$-Right Angled Artin Groups ($\Delta$-RAAGs) and show that the ones occuring as absolute Galois groups are exactly the ones which are absolute Galois groups of Formally real Pythagorean fields of finite type.
The last chapter concludes with a complete answer to a question from Min{\' a}{\v c}-Rogelstad-Tân for which Hamza had already given a partial answer for mild groups.