Field Theory

Intended for graduate courses or for independent study, this book presents the basic theory of fields. The first part begins with a discussion of polynomials over a ring, the division algorithm, irreducibility, field extensions, and embeddings. The second part is devoted to Galois theory. The third part of the book treats the theory of binomials. The book concludes with a chapter on families of binomials – the Kummer theory.

This new edition has been completely rewritten in order to improve the pedagogy and to make the text more accessible to graduate students. The exercises have also been improved and a new chapter on ordered fields has been included.

About the first edition:

" ...the author has gotten across many important ideas and results. This book should not only work well as a textbook for a beginning graduate course in field theory, but also for a student who wishes to take a field theory course as independent study."

-J.N. Mordeson, Zentralblatt

"The book is written in a clear and explanatory style. It contains over 235 exercises which provide a challenge to the reader. The book is recommended for a graduate course in field theory as well as for independent study."

- T. Albu, MathSciNet

Preliminaries.- Preliminaries.- Field Extensions.- Polynomials.- Field Extensions.- Embeddings and Separability.- Algebraic Independence.- Galois Theory.- Galois Theory I: An Historical Perspective.- Galois Theory II: The Theory.- Galois Theory III: The Galois Group of a Polynomial.- A Field Extension as a Vector Space.- Finite Fields I: Basic Properties.- Finite Fields II: Additional Properties.- The Roots of Unity.- Cyclic Extensions.- Solvable Extensions.- The Theory of Binomials.- Binomials.- Families of Binomials.