This book aims to provide thorough coverage of the main topics of abstract algebra while remaining accessible
to students with little or no previous exposure to abstract mathematics. It can be used either for a one-semester
introductory course on groups and rings or for a full-year course. More specifics on possible course plans using
the book are given in this preface.
Style of Presentation
Over many years of teaching abstract algebra to mixed groups of undergraduates, including mathematics majors,
mathematics education majors, and computer science majors, I have become increasingly aware of the difficulties
students encounter making their first acquaintance with abstract mathematics through the study of algebra. This
book, based on my lecture notes, incorporates the ideas I have developed over years of teaching experience on how
best to introduce students to mathematical rigor and abstraction while at the same time teaching them the basic
notions and results of modern algebra.
Two features of the teaching style I have found effective are repetition and especially an examples first, definitions
later order of presentation. In this book, as in my lecturing, the hard conceptual steps are always prepared for
by working out concrete examples first, before taking up rigorous definitions and abstract proofs. Absorption of
abstract concepts and arguments is always facilitated by first building up the student's intuition through experience
with specific cases.
Another principle that is adhered to consistently throughout the main body of the book (Parts A and B) is that
every algebraic theorem mentioned is given either with a complete proof, or with a proof broken up into to steps
that the student can easily fill in, without recourse to outside references. The book aims to provide a self-contained
treatment of the main topics of algebra, introducing them in such a way that the student can follow the arguments
of a proof without needing to turn to other works for help.
Throughout the book all the examples, definitions, and theorems are consecutively numbered in order to make
locating any particular item easier for the reader.
Coverage of Topics
In order to accommodate students of varying mathematical, backgrounds, an optional Chapter 0, at the beginning,
collects basic material used in the development of the main theories of algebra. Included are, among other topics,
equivalence relations, the binomial theorem, De Moivre's formula for complex numbers, and the fundamental theorem
of arithmetic. This chapter can be included as part of an introductory course or simply referred to as needed in
later chapters.
Special effort is made in Chapter 1 to introduce at the beginning all main types of groups the student will
be working with in later chapters. The first section of the chapter emphasizes the fact that concrete examples
of groups come from different sources, such as geometry, number theory, and the theory of equations.
Chapter 2 introduces the notion of group homomorphism first and then proceeds to the study of normal subgroups
and quotient groups. Studying the properties of the kernel of a homomorphism before introducing the definition
of a normal subgroup makes the latter notion less mysterious for the student and easier to absorb and appreciate.
A similar order of exposition is adopted in connection with rings. After the basic notion of a ring is introduced
in Chapter 6, Chapter 7 begins with ring homomorphisms, after which consideration of the properties of the kernels
of such homomorphisms gives rise naturally to the notion of an ideal in a ring.
Each chapter is designed around some central unifying theme. For instance, in Chapter 4 the concept of group
action is used to unify such results as Cayley's theorem, Burnside's counting formula, the simplicy of A5,
and the Sylow theorems and their applications.
The ring of polynomials over a field is the central topic of Part B, Rings and Fields, and is given a full chapter
of its own, Chapter 8. The traditional main topic in algebra, the solution of polynomial equations, is emphasized.
The solutions of cubics and quartics are introduced in Chapter 8. In Chapter 9 Euclidean domains and unique factorization
domains are studied, with a section devoted to the Gaussian integers. The fundamental theorem of algebra is stated
in Chapter 10. In Chapter 11 the connection among solutions of quadratic, cubic, and quartic polynomial equations
and geometric constructions is explored.
In Chapter 12, after Galois theory is developed, it is applied to give a deeper understanding of all these topics.
For instance, the possible Galois groups of cubic and quartic polynomials are fully worked out, and Artin's Galois-theoretic
proof of the fundamental- theorem of algebra, using nothing from analysis but the intermediate value theorem, is
presented. The chapter, and with it the main body of the book, culminates in the proof of the insolubility of the
general quintic and the construction of specific examples of quintics that are not solvable by radicals.
A brief history of algebra is given in Chapter 13, after Galois theory (which was the main historical source
of the group concept) has been treated, thus making a more meaningful discussion of the evolution of the subject
possible.
A collection of additional topics, several of them computational, is provided in Part C. In contrast to the
main body of the book (Parts A and B), where completeness is the goal, the aim in Part C is to give the student
an introduction to�and some taste of�a topic, after which a list of further references is provided for those who
wish to learn more. Instructors may include as much or as little of the material on a given topic as time and inclination
indicate.
Each chapter in the book is divided into sections, and each section provided with a set of exercises, beginning
with the more computational and proceeding to the more theoretical. Some of the theoretical exercises give a first
introduction to topics that will be treated in more detail later in the book, while others introduce supplementary
topic not otherwise covered, such as Cayley digraphs, formal power series, and the existence of transcendental
numbers.
Suggestions for Use
A one-semester introductory course on groups and rings might include Chapter 0 (optional); Chapters 1, 2, and
3 on groups; and Chapters 6, 7, and 8 on rings.
For a full-year course, Parts A and B, Chapters 1 through 12, offer a comprehensive treatment of the subject.
Chapter 9, on Euclidean domains, and Chapter 11, on geometric constructions, can be treated as optional supplementary
topics, depending on time arid the interest of the students and the instructor.
An instructor's manual, with solutions to all exercises plus further comments and suggestions, is available.
Instructors can obtain it by directly contacting the publisher, Prentice Hall.
Acknowledgments
It is a pleasure to acknowledge various contributors to the development of this book. First I should thank the
students of The College of New Jersey who have taken courses based on a first draft. I am grateful also to my colleagues
Andrew Clifford, Tom Hagedorn, and Dave Reimer for useful suggestions.
Special thanks are due to my colleague Ed Conjura, who taught from a craft of the book and made invaluable suggestions
for improvement that have been incorporated into the final version.
I am also most appreciative of the efforts of the anonymous referees engaged by the publisher, who provided
many helpful and
Summary
For a one-semester course covering groups and rings or a two-semester course in Abstract Algebra.
This text provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of
applications. A repetition and examples first approach introduces students to mathematical rigor and abstraction
while teaching them the basic notions and results of modern algebra.
Features
Concrete examples precede difficult concepts.
Builds students' intuition through experience with specific examples so that they may more readily absorb abstract
concepts.
Repetition of key steps in proofs.
Gives students many opportunities to learn the difficult steps in proofs by first introducing them in examples,
then in the actual proof of the theorem, and again in later examples.
Complete algebraic theorems and proofs.
Combines either a complete proof or a proof that students can complete with every algebraic theorem mentioned
so that students do not have to reference outside sources.
Self-contained treatment of all main topics.
Introduces the topics essential to abstract algebra in such a way that students can follow the arguments of
a proof without needing to turn to other works for help.
Unifying theme in each chapter.
Helps students draw conceptual parallels.
Solutions of cubics and quartics are given first.
Allows students to explore the connections between those solutions and geometric constructions and learn how
their Galois groups are calculated.
A brief history of algebra is discussed in a separate chapter.
Allows for a more meaningful discussion of the evolution of the subject.
Applications such as Symmetries, Grobner bases, Coding Theory, and Boolean algebras are provided with references.
Gives students the opportunity to learn more on their own, and instructors flexibility in topical coverage.
Chapter 0�Covers basic material used in the development of the main theories of algebra.
Allows instructors to teach this material in an introductory course. Gives more advanced students a useful
reference.
The interplay between algebra and geometry is emphasized.
Shows students how closely the two subjects are linked�¤especially useful for future high school teachers
who will be teaching both subjects.
64 sets of exercises�Containing an average of 20 exercises each.
Gives students ample opportunity to practice the concepts they just learned.
A large number of worked out examples.
Illustrates for the student how an abstract concept is used before its formal definition is stated.
Table of Contents
Preface.
0. Background.
A. Group Theory.
1. Groups.
2. Group Homomorphisms.
3. Direct Products and Abelian Groups.
4. Group Actions.
5. Composition Series.
B. Rings and Fields.
6. Rings.
7. Ring Homomorphisms.
8. Rings of Polynomials.
9. Euclidean Domains.
10. Field Theory.
11. Geometric Constructions.
12. Galois Theory.
13. Historical Notes.
