Dissociation des Extensions Algebriques de Corps par les Extensions Galoisiennes ou Galsimples non Galoisiennes

TL;DR

This thesis explores the structure of algebraic field extensions by constructing maximal Galois towers, establishing a Galois analogue of the Jordan-Holder theorem, and analyzing the conditions under which extensions admit such towers.

Contribution

It introduces the concept of galtowerable extensions and Galois composition towers, extending the analogy between group theory and field extension theory.

Findings

Finite galtowerable extensions admit Galois composition towers.

A Galois analogue of Jordan-Holder theorem is established for galtowerable extensions.

Not all finite separable extensions are galtowerable.

Abstract

The fundamental theorem of arithmetic factorizes any integer into a product of prime numbers. The Jordan-Holder theorem dissolves many groups by their normal series which can be refined into composition series. The main topic of this thesis is the dissociation of field extensions. We dissociate algebraic extensions by their intermediate fields to build a tower with the greatest possible number of Galois steps. We call "galtowerable" the extensions admitting a field tower all the steps of which are Galois extensions,i.e. a "Galois tower". Two Galois towers of the same galtowerable extension (finite or infinite) admit equivalent refinements. We discuss refinement of Galois towers to define "Galois composition towers". A field extension admits a Galois composition tower if and only if it is finite and galtowerable. For such an extension we obtain a Galois analogue of the Jordan-Holder theorem for groups. Any finite group admits a normal series, but there exist finite and separable extensions which are not galtowerable. Is it possible to pursue the dictionary between group theory and extension theory in this case ?