Classical Galois theory

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Millicent Gay Mastin (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Andrew Long

Abstract: The purpose of this thesis is to examine the correspondence between groups of automorphsims and fields and to prove the Fundamental Theorem of Galois Theory. The fields under consideration are infinite. Chapter I is devoted to the basic definitions and theorems needed throughout the paper. It has been assumed that the reader has a knowledge of the basic properties of groups, rings, integral domains, fields, and isomorphisms. Standard theorems have been stated without proofs but with a reference to a proof. In Chapter II, the basic properties of field extensions, considered as vector spaces, are investigated. Several theorems which characterize splitting fields, simple extensions, and separable extensions are proved. It is then shown that a finite, separable extension is a simple extension. The concepts of automorphism groups and fixed fields are introduced in Chapter III. This discussion concludes by showing that there is a one-to-one correspondence between closed subfields and closed subgroups of the group of automorphisms of a field. Chapter IV is devoted to a discussion which characterizes normal extensions. The chapter concludes by establishing an important relationship between normal extensions and splitting fields. The results of the preceding chapters are then used in Chapter V to prove the Fundamental Theorem.

Additional Information

Language: English
Date: 1972
Galois theory

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