Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Jan Minac

2nd Supervisor

Christian Maire

Affiliation

Université de Franche-Comté, Institut FEMTO-ST, Besançon France

Co-Supervisor

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.

Summary for Lay Audience

Galois theory was originally introduced by Evariste Galois, a young French mathematician, in the $19$-th century to characterise polynomial equations which are solvable by radicals, i.e. the solutions can be expressed by a formula involving only integers, $n$-th roots, additions and multiplications. For this purpose, Galois studied the group permutation of the solutions, which is called Galois group. He was one of the founder of modern Group theory.

Before Galois, it was already known that polynomial equations of degree less than $4$ are all solvable by radicals. Abel, Galois and Ruffini were able to exhibit, for every integer $n$ larger than $5$, polynomials of degree $n$ which are not solvable by radicals. As an application, some problems from Antiquity were solved: doubling the cube, trisecting the angle and characterising all polygons which are constructible with compasses and straightedges.

A more modern version of Galois' ideas focuses on studying all Galois groups of all possible equations together collecting them into one large infinite group called the "absolute Galois group". Due to its sheer size, this group is extremely difficult to understand and challenges many mathematicians even today. My work, together with my collaborators, aims to find new facts about this group.

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