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Introduction to Abstract Algebra

Jonathan D. H. Smith
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics 3
[Reviewed by
John Perry
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A common approach to teaching abstract algebra is to begin with an overview of number theory, then dive into an axiomatic view of group theory. The emphasis on groups is often such that many students enrolled in the one-semester course typically required for a math major never encounter rings, let alone fields or semigroups.

Another approach, not uncommon, is to begin with rings. Rather than give preference to groups because of their structural simplicity, these textbooks appeal to students’ previous experience with common rings such as integers, polynomials, and matrices. This approach appears in the text A First Course in Abstract Algebra: Rings, Groups, and Fields, by Anderson and Feil, also from CRC Press. The introduction to Anderson and Feil’s text includes univariate polynomials as well as some number theory, with the goal of motivating the student to study rings by highlighting common properties between integers and polynomials, such as factorization into irreducibles and division with uniqueness of remainder.

Dr. Smith states in the preface to Introduction to Abstract Algebra that his goal is

to present algebra as the main tool underlying discrete mathematics and the digital world, much as calculus was accepted as the main tool for continuous mathematics and the analog world.

To achieve this goal, he adopts a strategy that this reviewer has not seen before. After the usual introduction to number theory, the second chapter starts neither with groups nor with rings, but with functions. Why? To develop semigroups and monoids of functions. For examples of semigroups of functions, Smith uses the set of nondecreasing functions, the set of translations of the number line, and the set of computable functions.

After definining an isomorphism as a bijective map of two sets, not yet with any homomorphism structure, the second chapter concludes with a section describing groups of permutations as a special kind of monoid. Since permutations are merely bijective functions, chapter two performs the neat trick of using a single framework — that of functions, objects with which students in an abstract algebra class ought to be very familiar — to introduce the student first to semigroups and monoids, and finally to groups.

The third and shortest chapter delves into topics concerned with equivalence, introducing modular arithmetic and an isomorphism theorem for sets.

The fourth chapter provides a thorough generalization of semigroups, monoids, groups, and equivalence classes, developing to the theory of cosets before concluding with multiplication tables (Cayley tables). The groups of integers and matrices appear here, along with direct products of groups and the submonoid of stochastic matrices.

The fifth chapter introduces the homomorphism structure, and works its way to an isomorphism theorem for groups, and concludes with Cayley’s Theorem.

The remaining chapters treat rings, fields, factorization, modules, group actions, and quasigroups. Each chapter concludes with exercises, of course, and the exercises might be considered rather common for a course in algebra — granted, of course, that in the second chapter the exercises are all about functions. Unlike many texts, there is no discussion of Galois theory; given the author’s goal, this is appropriate, and helps keep the text at about half the size of many others available on the market.

What the reviewer most enjoyed about the text were the collection of Study Projects at the end of each chapter. These projects offer a large variety of explorations of the material, and give the text an applied flavor. A small sampling of the topics covered includes:

  1. improving on a bound for the number of steps required by the Euclidean Algorithm;
  2. showing that not all functions are computable;
  3. introducing cryptography and cryptanalysis;
  4. equivalence classes of musical frequencies;
  5. board games;
  6. check digits in bar codes; and
  7. discrete logarithms.

The study projects vary in level of difficulty, and include detailed background information along with a strategy for investigating the topic.

Smith includes some very limited biographical notes, but refrains from the common practice of giving sketches, explaining in the preface that more detailed information can be found in "libraries and the Internet."

One disappointment is that many of the study projects introduce topics that cry out for further development, but receive none. The most puzzling example is that a book whose aim is to show how algebra is the main mathematical tool underlying the digital world never mentions the contributions of algebra to public-key cryptography.

Moreover, the textbook adopts a few conventions that strike the reviewer as a little too unconventional. The most jarring was to see the permutation groups defined over the sets {0,1,…, n -1} rather the common {1,2,…, n }: for example, the text describes S 2 as the set {(0), (0 1)}.

Nevertheless, the reviewer’s overall impression is that the textbook’s approach succeeds admirably at developing a "third way" that balances the structural simplicity of starting with groups with the concrete familiarity of working in rings. It is more than adequate as the main textbook for two semesters of an undergraduate course, or as a supplemental resource. Those with a particular interest in applied and computational algebra may find it worthwhile as a resource for study projects, as would teachers at undergraduate institutions who would like to provide more concrete examples of algebra in the world. As such, the text provides a welcome new addition to the teaching of undergraduate algebra.

John Perry is an assistant professor of mathematics at the University of Southern Mississippi. His research interests lie in computational algebra. He enjoys classical novels, recently re-reading The Brothers Karamazov, and classical music, recently re-hearing Mozart’s Symphony no. 40 in D minor. He did not combine the two, but at certain points of the novel that might not be a bad idea.

Ordering Numbers
The Well-Ordering Principle
The Division Algorithm
Greatest Common Divisors
The Euclidean Algorithm
Primes and Irreducibles
The Fundamental Theorem of Arithmetic
Specifying Functions
Composite Functions
Linear Functions
Semigroups of Functions
Injectivity and Surjectivity
Groups of Permutations
Kernel and Equivalence Relations
Equivalence Classes
Rational Numbers
The First Isomorphism Theorem for Sets
Modular Arithmetic
Groups and Monoids 
Componentwise Structure
Submonoids and Subgroups
Multiplication Tables
Normal Subgroups
The First Isomorphism Theorem for Groups
The Law of Exponents
Cayley’s Theorem
Ring Homomorphisms
Quotient Rings
Polynomial Rings
Integral Domains
Polynomials over Fields
Principal Ideal Domains
Irreducible Polynomials
Lagrange Interpolation
Fields of Fractions
Factorization in Integral Domains
Noetherian Domains
Unique Factorization Domains
Roots of Polynomials
Splitting Fields
Uniqueness of Splitting Fields
Structure of Finite Fields
Galois Fields
Representing a Ring
Direct Sums
Free Modules
Vector Spaces
Abelian Groups
Group Actions 
Transitive Actions
Fixed Points
Faithful Actions
Alternating Groups
Sylow Theorems
Latin Squares
Quasigroup Homomorphisms
Quasigroup Homotopies
Principal Isotopy