Galois Theory: A Topological and Group Theoretic Construction

The definition of the Galois group of any field extension F/K is stated, and the fundamental theorem of Galois theory for algebraic Galois field extensions is proven. Further uncovering the structure of field extensions via their Galois groups, a definiton of the the Krull topology is given that makes the Galois group a topological group that is in fact a profinite group. Topological properties of Galois groups are studied along with the connections these properties have with the corresponding field extensions. A topological characterization of Galois groups of algebraic Galois extensions is then given. The group structure of Galois groups is analyzed via their dependence on and similarities to the structure of finite groups. Most of the main theorems are followed by examples where the theorem is applied to a specific case. The thesis is concluded with a brief discussion of analog Galois theories in other mathematical disciplines.