Cogalois theory versus galois theory

Abstract

ABSTRACT Since the middle of the sixteenth century to the beginning of the nineteenth cen tury some of the greatest mathematicians of this period have tried to obtain a formula for the roots of quintic equations. Before that period it was known that the solutions of the equations of degree less than or equal to 4 were expressible by radicals, in other words, the polynomials up to the quartic were solvable by radicals (a radical is a formula involving only the 4 basic arithmetic operations and the extraction of roots). Galois not only solved this important problem which could not be solved for centuries, but he also provided a criterion for solvability by radicals of any equation xn + an^ixn~l +... = 0. Actually, the importance of the main result of Galois' dis coveries has transcended by far that of the original problem which lead to it. His discoveries in the theory of equations is called Galois Theory. Galois Theory inves tigates field extensions possessing a Galois correspondence. Cogalois Theory, a fairly new theory of about 20 years old is dual to the Galois Theory and investigates field extensions possessing a Cogalois correspondence. The main objective of this work is to present the fundamentals of Galois Theory and Cogalois Theory. Firstly, we investigate Galois Theory. We start by providing some concepts in order to define the Galois extensions and the Galois group of a given field extension. Two major results of Galois Theory are given, namely The Funda mental Theorem of Finite Galois Theory and The Galois ' Criterion for Solvability by Radicals. Then we work on Cogalois Theory. In that part we provide some results, such as The Kneser Criterion and The Greither-Harrison Criterion. Lastly, we de fine G-Cogalois extensions. A G-Cogalois extension is a separable field extension with G/F*-Cogalois correspondence. The importance of these extensions stems from the fact that they play the same role as that of Galois extensions in Galois Theory. iv