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Publisher:

Chapman & Hall/CRC

Publication Date:

2009

Number of Pages:

327

Format:

Hardcover

Series:

Textbooks in Mathematics 3

Price:

89.95

ISBN:

9781420063714

Category:

Textbook

[Reviewed by , on ]

John Perry

02/25/2009

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:

- improving on a bound for the number of steps required by the Euclidean Algorithm;
- showing that not all functions are computable;
- introducing cryptography and cryptanalysis;
- equivalence classes of musical frequencies;
- board games;
- check digits in bar codes; and
- 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.

**Numbers**

Ordering Numbers

The Well-Ordering Principle

Divisibility

The Division Algorithm

Greatest Common Divisors

The Euclidean Algorithm

Primes and Irreducibles

The Fundamental Theorem of Arithmetic

**Functions**

Specifying Functions

Composite Functions

Linear Functions

Semigroups of Functions

Injectivity and Surjectivity

Isomorphisms

Groups of Permutations

**Equivalence**

Kernel and Equivalence Relations

Equivalence Classes

Rational Numbers

The First Isomorphism Theorem for Sets

Modular Arithmetic

**Groups and Monoids**

Semigroups

Monoids

Groups

Componentwise Structure

Powers

Submonoids and Subgroups

Cosets

Multiplication Tables

**Homomorphisms**

Homomorphisms

Normal Subgroups

Quotients

The First Isomorphism Theorem for Groups

The Law of Exponents

Cayley’s Theorem

**Rings**

Rings

Distributivity

Subrings

Ring Homomorphisms

Ideals

Quotient Rings

Polynomial Rings

Substitution

**Fields**

Integral Domains

Degrees

Fields

Polynomials over Fields

Principal Ideal Domains

Irreducible Polynomials

Lagrange Interpolation

Fields of Fractions

**Factorization**

Factorization in Integral Domains

Noetherian Domains

Unique Factorization Domains

Roots of Polynomials

Splitting Fields

Uniqueness of Splitting Fields

Structure of Finite Fields

Galois Fields

**Modules**

Endomorphisms

Representing a Ring

Modules

Submodules

Direct Sums

Free Modules

Vector Spaces

Abelian Groups

**Group Actions**

Actions

Orbits

Transitive Actions

Fixed Points

Faithful Actions

Cores

Alternating Groups

Sylow Theorems

**Quasigroups**

Quasigroups

Latin Squares

Division

Quasigroup Homomorphisms

Quasigroup Homotopies

Principal Isotopy

Loops

**Index**

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