Field Theory / Edition 2

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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."

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Editorial Reviews

From the Publisher

From the reviews of the second edition:

"Springer has just released the second edition of Steven Roman’s Field Theory, and it continues to be one of the best graduate-level introductions to the subject out there. … Every section of the book has a number of good exercises that would make this book excellent to use either as a textbook or to learn the material on your own. All in all, I recommend this book highly as it is a well-written expository account of a very exciting area in mathematics." (Darren Glass, The MAA Mathematical Sciences Digital Library, February, 2006)

“The second edition of Roman’s Field Theory … offers a graduate course on Galois theory. … The author’s approach is mainly standard … . the merits of such an approach would have been helpful for readers who already know some Galois theory, or for instructors who have to pick a textbook. … The clarity of exposition and lots of exercises make this a suitable textbook for a graduate course on Galois theory.” (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1172, 2009)

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Product Details

  • ISBN-13: 9780387276779
  • Publisher: Springer New York
  • Publication date: 11/17/2005
  • Series: Graduate Texts in Mathematics Series, #158
  • Edition description: 2nd ed. 2006
  • Edition number: 2
  • Pages: 335
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.81 (d)

Table of Contents

Pt. I Field extensions
1 Polynomials 23
2 Field extensions 41
3 Embeddings and separability 73
4 Algebraic independence 93
Pt. II Galois theory
5 Galois theory I : an historical perspective 113
6 Galois theory II : the theory 137
7 Galois theory III : the Galios group of a polynomial 173
8 A field extension as a vector space 197
9 Finite fields I : basic properties 211
10 Finite fields II : additional properties 225
11 The roots of unity 239
12 Cyclic extensions 261
13 Solvable extensions 269
Pt. III The theory of binomials
14 Binomials 289
15 Families of binomials 309
App Mobius inversion 319
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