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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."
This new edition has been completely rewritten for greater accessibility, and includes considerable new material The new chapter on ordered sets contains material that cannot be found in other books The exercises have also been improved for the new edition Intended for graduate courses or for independent study
Klappentext
This book presents the basic theory of fields, starting more or less from the beginning. It is suitable for a graduate course in field theory, or independent study. The reader is expected to have taken an undergraduate course in abstract algebra, not so much for the material it contains but in order to gain a certain level of mathematical maturity.
For this new edition, the author has rewritten the text based on his experiences teaching from the first edition. There are new exercises, a new chapter on Galois theory from an historical perspective, and additional topics sprinkled throughout the text, including a proof of the Fundamental Theorem of Algebra, a discussion of casus irreducibilis, Berlekamp's algorithm for factoring polynomials over Zp and natural and accessory irrationalities.
From the reviews of the first edition:
The book is written in a clear and explanatory style...the book is recommended for a graduate course in field theory as well as for independent study.
...[the author] does an excellent job of stressing the key ideas. 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.
Inhalt
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.