Abstract Algebra: Theory and Applications

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Thomas Judson, Stephen F. Austin State University

Pub Date: 2016

ISBN 13: 978-1-9443250-2-2

Publisher: University of Puget Sound

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Reviewed by Daniel Hernández, Assistant Professor, University of Kansas, on 8/22/2016.

This book is introductory, and covers the basic of groups, rings, fields, and vector spaces. In addition, it also includes material on some … read more

 

Table of Contents

  1. Preliminaries
  2. The Integers
  3. Groups
  4. Cyclic Groups
  5. Permutation Groups
  6. Cosets and Lagrange’s Theorem
  7. Introduction to Cryptography
  8. Algebraic Coding Theory
  9. Isomorphisms
  10. Normal Subgroups and Factor Groups
  11. Homomorphisms
  12. Matrix Groups and Symmetry
  13. The Structure of Groups
  14. Group Actions
  15. The Sylow Theorems
  16. Rings
  17. Polynomials
  18. Integral Domains
  19. Lattices and Boolean Algebras
  20. Vector Spaces
  21. Fields
  22. Finite Fields
  23. Galois Theory

About the Book

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation.

This text contains more material than can possibly be covered in a single semester. Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text. The order of presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21. Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor. A two-semester course emphasizing theory might cover Chapters 1 through 6, 9, 10, 11, 13 through 18, 20, 21, 22 (the first part), and 23. On the other hand, if applications are to be emphasized, the course might cover Chapters 1 through 14, and 16 through 22. In an applied course, some of the more theoretical results could be assumed or omitted. A chapter dependency chart appears below. (A broken line indicates a partial dependency.)

About the Contributors

Author(s)

Thomas W. Judson, Associate Professor, Department of Mathematics and Statistics, Stephen F. Austin State University. PhD University of Oregon.