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Abstract Algebra: Structures and Applications

Stephen Lovett
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2015
Number of Pages: 
720
Format: 
Hardcover
Price: 
89.95
ISBN: 
9781482248906
Category: 
Textbook
[Reviewed by
Mark Hunacek
, on
11/30/2015
]

This is a text for a serious upper-level undergraduate course in abstract algebra. It adopts a “groups first” approach to the subject, and, although it starts from scratch, winds up covering more than enough material to fill out two semesters. The topic coverage is very extensive for an undergraduate text: all the standard topics that one might want to cover in a two-semester introduction to the subject appear here, as do, to varying degrees, several topics (e.g., modules, path algebras, group representation theory, an introduction to algebraic geometry, categories) that seem quite unlikely to wind up getting taught to undergraduates.

The first five chapters roughly correspond to what I generally cover in a first-semester abstract algebra course, though there is more here than I can ever get to. The book starts with a discussion of the basics (sets and functions, equivalence relations, partial orderings), followed by a chapter on basic number theory (divisibility, congruences and modular arithmetic, mathematical induction); after this, there are two chapters on introductory group theory (up through normal subgroups, quotient groups and the Fundamental Theorem of Finitely Generated Groups), followed by one on the beginnings of ring theory (from the definition through quotient rings, and maximal and prime ideals). As in most undergraduate texts, but not most upper-level ring theory books, the definition of a ring does not presuppose a multiplicative identity.

In my first-semester abstract algebra courses, I can usually cover most or all of the group theory mentioned above (though I have to content myself with a statement, without proof, of the Fundamental Theorem of Finite Abelian Groups, and do not talk about finitely generated groups) and some of the ring theory (I can sometimes manage to get to ideals, but rarely have time to do maximal and prime ones). Interestingly, though, the author states in the preface that he views the first seven chapters as being suitable for a first semester course.

This seems quite ambitious to me: chapters 6 and 7 are on factorization theory in commutative rings (unique factorization domains, Euclidean domains, algebraic integers) and field extensions (including splitting fields, algebraic closure, and finite fields). Even the basic material on these subjects is, to my knowledge, never covered in our first semester algebra course (and certainly never covered by me); the material in these chapters, however, goes well beyond the basics. For example, the text goes into greater detail than is customary on the solutions of cubics and quadrics, gives a proof of Liouville’s theorem on approximation of algebraic numbers (which is used to prove certain numbers are not algebraic), and goes into a great deal of geometric detail on the subject of constructible numbers.

The remainder of the text (comprising chapters 8–13) covers additional advanced topics in ring and field theory (modules, including modules over a PID with applications to linear algebra; a fairly detailed look at Galois theory; an introduction to algebraic geometry, including Gröbner bases and their applications), group theory (group actions; Sylow theory; an introduction to group representation theory, including a proof of Maschke’s theorem; solvable and nilpotent groups), and, in a final chapter, categories and functors.

The author states that these chapters would be appropriate for a second semester course, but again there is no doubt in my mind that I couldn’t get through anywhere near all of this material in a second semester. (Nor would I want to: I think there is a time and place to teach categories and functors, and a second-semester undergraduate course in algebra is neither.) In such a course I would be happy to do some advanced group theory (group actions and Sylow theory seem a natural topic), followed by an introduction to rings, including an introduction to field extensions and perhaps some hand-waving in the direction of Galois theory at the very end of the course. It would never occur to me, in a second semester undergraduate algebra course, to talk about some of the more sophisticated topics referred to above.

This is, therefore, not a book for the faint of heart, although the author does do quite a good job of making this material accessible. The explanations are clear and are accompanied by lots of examples. Exercises are very plentiful, and cover a wide range of difficulty; there are quite a few that would, I think, challenge an average student (and some that might challenge an average professor). Some were new to me, and one, in fact, has already proved to be quite welcome: the evening before I was to prove in my geometry class that a regular heptagon is not constructible with compass and straightedge (a proof that, in past incarnations of the course, has always taken a considerable amount of class time, because I had to develop the material on complex numbers that I needed), I read an exercise in the book that leads the reader through a proof that uses only basic trigonometry, and allowed me to give a much simpler and quicker presentation in class.

The exercises are not arranged in increasing order of difficulty. For example, in section 5.1, which introduces rings, exercise 37 is the famous problem “if x3 = x for all x in a ring R, then R is commutative.” This is a hard problem; I spent months, off and on, thinking about it as an undergraduate after noticing it as an exercise in Herstein’s Topics in Algebra. It is followed in the book under review by the vastly simpler exercise 46, which merely asks the reader to show by example that the union of two subrings need not be a subring. Nor does the author use any kind of symbol (asterisk, dagger, etc.) to flag the exercises that he believes to be of above-average difficulty. This is not a problem in a classroom, where a professional is available to pick and choose the problems for the class, but may prove discouraging to students using this book for self-study.

In addition to exercises, each chapter ends with a list of Projects. These tend to be quite demanding and often require the student to write a paper or computer algorithm. Some require the use of a computer algebra system. (Knowledge of a computer algebra system is not a prerequisite for the text, however.) These Projects introduce a host of interesting side issues, including: fuzzy sets, symmetry of Sudoku puzzles, results on matrix groups, quaternion matrices, quotient rings in calculus, and much, much more.

The author does an excellent job of balancing theory with applications. While theory is certainly developed at some length, there are (in addition to the Projects) a number of discussions of nontrivial applications, some (such as the Diffie-Hellman Key Exchange and RSA cryptography) lengthy enough to merit an entire section.

One other point should be noted. The CRC Press webpage for the book describes this text as a “discovery-based approach”. To me, this phrase brings to mind a “Moore method”-style text, in which the student is led to discover, for himself or herself, the details of proofs or solutions to problems. (See, e.g., the review in this column of Gay’s Explorations in Topology.) This is not that sort of book, however. Theorems are, for the most part, proved in the text in the traditional way, which I think is a good thing: I have not yet bought into the prove-it-yourself philosophy.

The inclusion of all the topics described above, and the large number of exercises, examples and Projects, make for an undeniably interesting text, but there is a downside: this book is not only mathematically heavy, but physically heavy as well. CRC Press seems to like publishing large books, but even by their standards this one stands out. It is almost a foot tall, very thick, and appears to me to weigh at least five pounds. Contrast this with Dummit and Foote’s Abstract Algebra, which at 932 pages is more than 200 pages longer than this text, yet doesn’t seem to be any thicker, is about two inches less tall and about an inch less long. Lovett’s book is so unwieldy that I found it awkward to read, and I certainly can’t imagine a student dragging it back and forth (which many students like to do, if only to get an early start between classes on homework assignments taken from the text).

So, to summarize: this is a well-written book with interesting features, but its sheer volume and weight would give me serious pause before using it as a text. At least based on my own experience teaching abstract algebra at Iowa State, I would think that most undergraduates would find it somewhat overwhelming, both physically and mentally. Its “highest and best use”, as real estate appraisers say, seems to me to be as a reference for students, or faculty members teaching a course in abstract algebra.


Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.

SET THEORY
Sets and Functions
The Cartesian Product; Operations; Relations
Equivalence Relations
Partial Orders

 

NUMBER THEORY
Basic Properties of Integers
Modular Arithmetic
Mathematical Induction

 

GROUPS
Symmetries of the Regular n-gon
Introduction to Groups
Properties of Group Elements
Symmetric Groups
Subgroups
Lattice of Subgroups
Group Homomorphisms
Group Presentations
Groups in Geometry
Diffie-Hellman Public Key
Semigroups and Monoids

 

QUOTIENT GROUPS
Cosets and Lagrange’s Theorem
Conjugacy and Normal Subgroups
Quotient Groups
Isomorphism Theorems
Fundamental Theorem of Finitely Generated Abelian Groups

 

RINGS
Introduction to Rings
Rings Generated by Elements
Matrix Rings
Ring Homomorphisms
Ideals
Quotient Rings
Maximal and Prime Ideals

 

DIVISIBILITY IN COMMUTATIVE RINGS
Divisibility in Commutative Rings
Rings of Fractions
Euclidean Domains
Unique Factorization Domains
Factorization of Polynomials
RSA Cryptography
Algebraic Integers

 

FIELD EXTENSIONS
Introduction to Field Extensions
Algebraic Extensions
Solving Cubic and Quartic Equations
Constructible Numbers
Cyclotomic Extensions
Splitting Fields and Algebraic Closures
Finite Fields

 

GROUP ACTIONS
Introduction to Group Actions
Orbits and Stabilizers
Transitive Group Actions
Groups Acting on Themselves
Sylow’s Theorem
A Brief Introduction to Representations of Groups

 

CLASSIFICATION OF GROUPS
Composition Series and Solvable Groups
Finite Simple Groups
Semidirect Product. Classification Theorems
Nilpotent Groups

 

MODULES AND ALGEBRAS
Boolean Algebras
Vector Spaces
Introduction to Modules
Homomorphisms and Quotient Modules
Free Modules and Module Decomposition
Finitely Generated Modules over PIDs, I
Finitely Generated Modules over PIDs, II
Applications to Linear Transformations
Jordan Canonical Form
Applications of the Jordan Canonical Form
A Brief Introduction to Path Algebras

 

GALOIS THEORY
Automorphisms of Field Extensions
Fundamental Theorem of Galois Theory
First Applications of Galois Theory
Galois Groups of Cyclotomic Extensions
Symmetries among Roots; The Discriminant
Computing Galois Groups of Polynomials
Fields of Finite Characteristic
Solvability by Radicals

 

MULTIVARIABLE POLYNOMIAL RINGS
Introduction to Noetherian Rings
Multivariable Polynomial Rings and Affine Space
The Nullstellensatz
Polynomial Division; Monomial Orders
Gröbner Bases
Buchberger’s Algorithm
Applications of Gröbner Bases
A Brief Introduction to Algebraic Geometry

 

CATEGORIES
Introduction to Categories
Functors

 

APPENDICES

LIST OF NOTATIONS

BIBLIOGRAPHY

INDEX

Dummy View - NOT TO BE DELETED