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Publisher:

Open source

Publication Date:

2011

Number of Pages:

438

Format:

Electronic Book

Price:

0.00

ISBN:

open source book

Category:

Textbook

[Reviewed by , on ]

Christopher P. Thron

10/20/2011

For many students, abstract algebra is the most daunting of math classes. Many students (particularly those who do not have a strong theoretical bent) see abstract algebra as symbol-twiddling with no apparent rhyme or reason. To them, group theory proofs are just so many rabbits pulled from hats.

In our math department at Texas A&M — Central Texas, most of the majors are certifying secondary teachers. For this reason, I needed a textbook that made heavy use of familiar mathematical structures such as the integers mod n, complex numbers, symmetries, and permutations to motivate and illustrate the more abstract concepts. I also needed a more pedestrian book that valued clarity and gradual unfolding above elegance and conciseness. The book’s presentation should be interspersed with numerous, easily-worked examples. The exercises should be progressive, with a generous number of relatively easy problems for student practice. Practical applications of abstract algebra should figure prominently.

Of all the prospective texts I looked at from the standpoint of these requirements, Thomas Judson’s *Abstract Algebra: Theory and Applications* (*AATA*) was the best. (The fact that it was free was an added bonus.) The level was non-threatening, and the order and presentation of topics seemed perfect for what I was looking for. The “Preliminaries” chapter begins with several pointers on reading and writing proofs — vital background knowledge that most a abstract algebra books take for granted. Next, the book covers sets and equivalence relations in a way that bridges from familiar material to a more abstract setting. In the chapters dealing with groups, there are entire sections devoted to the integers mod n, symmetries, and complex numbers. Some practical topics such as ISBN and UPC codes are well-covered in the exercises; while others such as cryptography (the discussion of RSA is a bit brief) and algebraic coding (group codes, linear codes, and polynomial codes) are treated in well-placed optional chapters.

Despite these enabling features, my students in their end-of-semester evaluations commented that they wanted even more exposition and even more examples to bridge to the problems. These reactions are probably more a reflection of deficiencies in the students’ backgrounds than of deficiencies in AATA. Nonetheless, their comments led me to exploit one of the greatest strengths of AATA: full customizability. All of the LaTeX code is freely available online. My graduate student Justin Hill and I took Judson’s source material and in a single semester developed a book that was precisely suited to the background and interests of our particularly students. This turns on its head the conventional model of textbook development, which requires slow evolution of lecture notes over the course of several semesters.

*Abstract Algebra: Theory and Applications* is open-source in the fullest sense of the word. The source code is kept in a repository under version control and textbook adopters are encouraged to submit changes. A new edition has been put out every year for the past three years — all editions and the repository may be accessed from the download page.

There were several additional attractive features of *AATA* that I did not take advantage of. The book has sufficient material for a complete two-semester course covering groups, rings, and fields. There is also an accompanying SAGE workbook by Rob Beezer that supports the text (SAGE is an open-source software package that does abstract algebra, including operations with finite groups, polynomial rings, finite fields, field extensions, and more.)

In short, AATA is a stellar example of open-source at its best. The craftsmanship is top-notch, and is being continuously improved. I believe this book makes a strong case for open-source textbooks, and is in the vanguard of a revolution which will completely change the way future textbooks are developed, adapted, and utilized.

Chris Thron obtained his Ph.D. from the University of Wisconsin, taught in China and at King College (Bristol TN), worked for 10 years as a systems engineer at Freescale Semiconductor, and is now assistant professor of math at Texas A&M University — Central Texas.

Preface

1 Preliminaries

1.1 A Short Note on Proofs

1.2 Sets and Equivalence Relations

2.1 Mathematical Induction

2.2 The Division Algorithm

3 Groups

3.1 Integer Equivalence Classes and Symmetries

3.2 Definitions and Examples

3.3 Subgroups

4 Cyclic Groups

4.1 Cyclic Subgroups

4.2 Multiplicative Group of Complex Numbers

4.3 The Method of Repeated Squares

5 Permutation Groups

5.1 Definitions and Notation

5.2 Dihedral Groups

6 Cosets and Lagrange’s Theorem

6.1 Cosets

6.2 Lagrange’s Theorem

6.3 Fermat’s and Euler’s Theorems

7 Introduction to Cryptography

7.1 Private Key Cryptography

7.2 Public Key Cryptography

8 Algebraic Coding Theory

8.1 Error-Detecting and Correcting Codes

8.2 Linear Codes

8.3 Parity-Check and Generator Matrices

8.4 Efficient Decoding

9 Isomorphisms

9.1 Definition and Examples

9.2 Direct Products

10 Normal Subgroups and Factor Groups

10.1 Factor Groups and Normal Subgroups

10.2 The Simplicity of the Alternating Group

11 Homomorphisms

11.1 Group Homomorphisms

11.2 The Isomorphism Theorems

12 Matrix Groups and Symmetry

12.1 Matrix Groups

12.2 Symmetry

13 The Structure of Groups

13.1 Finite Abelian Groups

13.2 Solvable Groups

14 Group Actions

14.1 Groups Acting on Sets

14.2 The Class Equation

14.3 Burnside’s Counting Theorem

15 The Sylow Theorems

15.1 The Sylow Theorems

15.2 Examples and Applications

16 Rings

16.1 Rings

16.2 Integral Domains and Fields

16.3 Ring Homomorphisms and Ideals

16.4 Maximal and Prime Ideals

16.5 An Application to Software Design

17 Polynomials

17.1 Polynomial Rings

17.2 The Division Algorithm

17.3 Irreducible Polynomials

18 Integral Domains

18.1 Fields of Fractions

18.2 Factorization in Integral Domains

19 Lattices and Boolean Algebras

19.1 Lattices

19.2 Boolean Algebras

19.3 The Algebra of Electrical Circuits

20 Vector Spaces

20.1 Definitions and Examples

20.2 Subspaces

20.3 Linear Independence

21 Fields

21.1 Extension Fields

21.2 Splitting Fields

21.3 Geometric Constructions

22 Finite Fields

22.1 Structure of a Finite Field

22.2 Polynomial Codes

23 Galois Theory

23.1 Field Automorphisms

23.2 The Fundamental Theorem

23.3 Applications

Hints and Solutions

Notation

Index

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