Fields and Galois Theory

This book gives a concise exposition of the theory of fields, including the Galois theory of field extensions, the Galois theory of etale algebras, and the theory of transcendental extensions. The first five chapters treat the material covered in most courses in Galois theory while the final four are more advanced.

The first two chapters are concerned with preliminaries on polynomials and field extensions, and Chapter 3 proves the fundamental theorems in the Galois theory of fields. Chapter 4 explains, with copious examples, how to compute Galois groups, and Chapter 5 describes the many applications of Galois theory.

In Chapter 6, a weak form of the Axiom of Choice is used to show that all fields admit algebraic closures, and that any two are isomorphic. The last three chapters extend Galois theory to infinite field extensions, to etale algebras over fields, and to nonalgebraic extensions.

The approach to Galois theory in Chapter 3 is that of Emil Artin, and in Chapter 8 it is that of Alexander Grothendieck.

This book originated as the notes for a first-year graduate course taught at the University of Michigan, but they have since been revised and expanded numerous times. The only prerequisites are an undergraduate course in abstract algebra and some group theory. There are 96 exercises, most with solutions.