Galois Theory and its Application to the Problem of Solvability by Radicals of an Equation Over a Field of Prime or Zero Characteristic

Abstract

In Part I of the thesis an account is given of the basic algebra of extension fields which is required for the understanding of Galois theory. The fundamental theorem states the relationships of the subgroups of a permutation group of the root field of an equation to the subfields which are left invariant by these subgroups. Extensions of the basic theorem conclude Part I. In part II the solvability of equations by radicals is discussed, for fields of characteristic zero. A discussion of finite fields and primitive roots leads to a criterion for the solvability by radicals of equations over fields of prime characteristic. Finally, a method for determining the Galois group of any equation is discussed. Most of the material in the introductory chapters is taken from Artin's: Galois Theory (cf. p. 120).