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

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 ?