ISBN 1-57766-443-4, © 2005, 484 pages, $73.95 list
Waveland Press, Inc.,
Click here for information about the Second Edition, including the appropriate Study Guide.
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Northern Illinois University,
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Some of the strengths of this undergraduate/graduate level textbook are
the gentle introduction to proof in a concrete setting,
the introduction of abstract concepts only after
a careful study of important examples,
and the gradual increase of the level of sophistication
as the student progresses through the book.
A number theory thread runs throughout several optional sections,
and there is an overview of techniques for computing Galois groups.
The book offers an extensive set of exercises
that help to build skills in writing proofs.
Chapter introductions, together with notes at the ends of certain chapters,
provide motivation and historical context,
while relating the subject matter to the broader mathematical picture.
TABLE OF CONTENTS
PREFACE | PREFACE TO THE SECOND EDITION | TO THE STUDENT | WRITING PROOFS
| HISTORICAL BACKGROUND
Chapter 1: INTEGERS | Divisors | Primes | Congruences | Integers Modulo n | Notes || 46 pages
Chapter 2: FUNCTIONS | Functions | Equivalence Relations | Permutations | Notes || 40 pages
Chapter 3: GROUPS | Definition of a Group | Subgroups | Constructing Examples | Isomorphisms | Cyclic Groups | Permutation Groups | Homomorphisms | Cosets, Normal Subgroups, and Factor Groups | Notes || 92 pages
Chapter 4: POLYNOMIALS | Fields; Roots of Polynomials | Factors | Polynomials with Integer Coefficients | Existence of Roots | Notes || 44 pages
Chapter 5: COMMUTATIVE RINGS | Commutative Rings; Integral Domains | Ring Homomorphisms | Ideals and Factor Rings | Quotient Fields | Notes || 46 pages
Chapter 6: FIELDS | Algebraic Elements | Finite and Algebraic Extensions | Geometric Constructions | Splitting Fields | Finite Fields | Irreducible Polynomials over Finite Fields | Quadratic Reciprocity | Notes || 46 pages
Chapter 7: STRUCTURE OF GROUPS | Isomorphism Theorems; Automorphisms | Conjugacy | Groups Acting on Sets | The Sylow Theorems | Finite Abelian Groups | Solvable Groups | Simple Groups || 50 pages
Chapter 8: GALOIS THEORY | The Galois Group of a Polynomial | Multiplicity of Roots | The Fundamental Theorem of Galois Theory | Solvability by Radicals | Cyclotomic Polynomials | Computing Galois Groups || 42 pages
Chapter 9: UNIQUE FACTORIZATION | Principal Ideal Domains | Unique Factorization Domains | Some Diophantine Equations || 26 pages
APPENDIX | Sets | Construction of the Number Systems | Basic Properties of the Integers | Induction | Complex Numbers | Solution of Cubic and Quartic Equations | Dimension of a Vector Space || 28 pages
BIBLIOGRAPHY | SELECTED ANSWERS | INDEX OF SYMBOLS | INDEX
484 pages, approximately 800 exercises
This edition would probably not have been written without the impetus from George Bergman, of the University of California, Berkeley. After using the book, on more than one occasion he sent us a large number of detailed suggestions on how to improve the presentation. Many of these were in response to questions from his students, so we owe an enormous debt of gratitude to his students, as well as to Professor Bergman. We believe that our responses to his suggestions and corrections have measurably improved the book.
We would also like to acknowledge important corrections and suggestions that we received from Marie Vitulli, of the University of Oregon, and from David Doster, of Choate Rosemary Hall. We have also benefitted over the years from numerous comments from our own students and from a variety of colleagues. We would like to add Doug Bowman, Dave Rusin, and Jeff Thunder to the list of colleagues given in the preface to the second edition.
In this edition we have added about 150 exercises, we have added 1 to all rings, and we have done our best to weed out various errors and misprints.
We use the book in a linear fashion, but there are some alternatives to that approach. With students who already have some acquaintance with the material in Chapters 1 and 2, it would be possible to begin with Chapter 3, on groups, using the first two chapters for a reference. We view these chapters as studying cyclic groups and permutation groups, respectively. Since Chapter 7 continues the development of group theory, it is possible to go directly from Chapter 3 to Chapter 7.
Chapter 5 contains basic facts about commutative rings, and contains many examples which depend on a knowledge of polynomial rings from Chapter 4. Chapter 5 also depends on Chapter 3, since we make use of facts about groups in the development of ring theory, particularly in Section 5.3 on factor rings. After covering Chapter 5, it is possible to go directly to Chapter 9, which has more ring theory and some applications to number theory.
Our development of Galois theory in Chapter 8 depends on results from Chapters 5 and 6. Section 8.4, on solvability by radicals, requires a significant amount of material from Chapter 7.
Rather than outlining a large number of possible paths through various parts of the text, we have to ask the instructor to read ahead and use a great deal of caution in choosing any paths other than the ones we have suggested above.
We would like to point out to both students and instructors that there is some supplementary material available on the book's website.
Finally, we would like to thank our publisher, Neil Rowe, for his continued support of our writing.
John A. Beachy
William D. Blair
September 1, 2005
As a prerequisite to the abstract algebra course, our students are required to have taken a sophomore level course in linear algebra that is largely computational, although they have been introduced to proofs to some extent. Our classes include students preparing to teach high school, but almost no computer science or engineering students. We certainly do not assume that all of our students will go on to graduate school in pure mathematics.
In searching for appropriate text books, we have found several texts that start at about the same level as we do, but most of these stay at that level, and they do not teach nearly as much mathematics as we desire. On the other hand, there are several fine books that start and finish at the level of our Chapters 3 through 6, but these books tend to begin immediately with the abstract notion of group (or ring), and then leave the average student at the starting gate. We have in the past used such books, supplemented by our Chapter 1.
Historically the subject of abstract algebra arose from concrete problems, and it is our feeling that by beginning with such concrete problems we will be able to generate the student's interest in the subject and at the same time build on the foundation with which the student feels comfortable.
Although the book starts in a very concrete fashion, we increase the level of sophistication as the book progresses, and, by the end of Chapter 6, all of the topics taught in our course have been covered. It is our conviction that the level of sophistication should increase, slowly at first, as the students become familiar with the subject. We think our ordering of the topics speaks directly to this assertion.
Recently there has been a tendency to yield to demands of ``relevancy,'' and to include ``applications'' in this course. It is our feeling that such inclusions often tend to be superficial. In order to make room for the inclusion of applications, some important mathematical concepts have to be sacrificed. It is clear that one must have substantial experience with abstract algebra before any genuine applications can be treated. For this reason we feel that the most honest introduction concentrates on the algebra. One of the reasons frequently given for treating applications is that they motivate the student. We prefer to motivate the subject with concrete problems from areas that the students have previously encountered, namely, the integers and polynomials over the real numbers.
One problem with most treatments of abstract algebra, whether they begin with group theory or ring theory, is that the students simultaneously encounter for the first time both abstract mathematics and the requirement that they produce proofs of their own devising. By taking a more concrete approach than is usual, we hope to separate these two initiations.
In three of the first four chapters of our book we discuss familiar concrete mathematics: number theory, functions and permutations, and polynomials. Although the objects of study are concrete, and most are familiar, we cover quite a few nontrivial ideas and at the same time introduce the student to the subtle ideas of mathematical proof. (At Northern Illinois University, this course and Advanced Calculus are the traditional places for students to learn how to write proofs.) After studying Chapters 1 and 2, the students have at their disposal some of the most important examples of groups-permutation groups, the group of integers modulo n, and certain matrix groups. In Chapter 3 the abstract definition of a group is introduced, and the students encounter the notion of a group armed with a variety of concrete examples.
Probably the most difficult notion in elementary group theory is that of a factor group. Again this is a case where the difficulty arises because there are, in fact, two new ideas encountered together. We have tried to separate these by treating the notions of equivalence relation and partition in Chapter 2 in the context of sets and functions. We consider there the concept of factoring a function into ``better'' functions, and show how the notion of a partition arises in this context. These ideas are related to the integers modulo n, studied in Chapter 1. When factor groups are introduced in Chapter 3, we have partitions and equivalence relations at our disposal, and we are able to concentrate on the group structure introduced on the equivalence classes.
In Chapter 4 we return to a more concrete subject when we derive some important properties of polynomials. Here we draw heavily on the students' familiarity with polynomials from high school algebra and on the parallel between the properties of the integers studied in Chapter 1 and the polynomials. Chapter 5 then introduces the abstract definition of a ring after we have already encountered several important examples of rings: the integers, the integers modulo n, and the ring of polynomials with coefficients in any field.
From this point on our book looks more like a traditional abstract algebra textbook. After rings we consider fields, and we include a discussion of root adjunction as well as the three problems from antiquity: squaring the circle, duplicating the cube, and trisecting an angle. We also discuss splitting fields and finite fields here. We feel that the first six chapters represent the most that students at institutions such as ours can reasonably absorb in a year.
Chapter 7 returns to group theory to consider several more sophisticated ideas including those needed for Galois theory, which is the subject matter of Chapter 8. In Chapter 9 we return to a study of rings, and consider questions of unique factorization. As a number theoretic application, we present a proof of Fermat's last theorem for the exponent 3. In fact, this is the last of a thread of number theoretic applications that run through the text, including a proof of the quadratic reciprocity law in Section 6.7 and a study of primitive roots modulo p in Section 7.5. The applications to number theory provide topics suitable for honors students.
The last three chapters are intended to make the book suitable for an honors course or for classes of especially talented or well-prepared students. In these chapters the writing style is rather terse and demanding. Proofs are included for the Sylow theorems, the structure theorem for finite abelian groups, theorems on the simplicity of the alternating group and the special linear group over a finite field, the fundamental theorem of Galois theory, Abel's theorem on the insolvability of the quintic, and the theorem that a polynomial ring over a unique factorization domain is again a unique factorization domain.
The only prerequisite for our text is a sophomore level course in linear algebra. We do not assume that the student has been required to write, or even read, proofs before taking our course. We do use examples from matrix algebra in our discussion of group theory, and we draw on the computational techniques learned in the linear algebra course-see, for example, our treatment of the Euclidean algorithm in Chapter 1.
We have included a number of appendices to which the student may be referred for background material. The appendices on induction and on the complex numbers might be appropriate to cover in class, and so they include some exercises.
In our classes we usually intend to cover Chapters 1, 2 and 3 in the first semester, and most of Chapters 4, 5 and 6 in the second semester. In practice, we usually begin the second semester with group homomorphisms and factor groups, and end with geometric constructions. We have rarely had time to cover splitting fields and finite fields. For students with better preparation, Chapters 1 and 2 could be covered more quickly. The development is arranged so that Chapter 7 on the structure of groups can be covered immediately after Chapter 3. On the other hand, the material from Chapter 7 is not really needed until Section 8.4, at which point we need results on solvable groups.
We have included answers to some of the odd numbered computational exercises.
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