Volume no :6, Issue no: 1, -2 September and December 2011

Journal of Algebra, Number Theory: Advances and Applications

TRANSLATION INTO GALOIS THEORY OF THEOREMS IN GROUP THEORY

Author's: EMMANUEL ANDRÉO and RICHARD MASSY
Pages: [29] - [56]
Received Date: October 13, 2011
Submitted by:

Abstract

We develop the theory of algebraic field extensions in a way similar to normal series of subgroups in group theory. Is it possible, by dissociating algebraic extensions through their intermediate fields, to “approximate” them by Galois extensions, in order to construct a tower with as many Galois steps as possible? We describe the obstruction to this Galois dissociation by proving a fundamental difference between groups and extensions: Every finite group admits a normal series, whereas a finite extension, even if separable, does not necessarily admit a Galois tower. For those extensions that are analogous in nature to groups, we establish a complete dictionary between groups and extensions by giving the Galois analogues of the most famous classical results of group theory.

Keywords

field towers, refinements, composition Galois towers, semi-abelian extensions, solvable extensions.