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Abstract Algebra: Theory and Applications

Thomas W. Judson
Open source
Publication Date: 
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Electronic Book
open source book
[Reviewed by
Christopher P. Thron
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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.


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

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