Introduction

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.

New to the fourth edition is a supplement “Selected Solutions for Students” that provides complete solutions to a number of important exercises in the text.

HOW WE HAVE USED THE BOOK

Our students have taken a linear algebra course, in which they have been exposed to some proofs, but they have not taken a “transition to higher mathematics” course.

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 two semester sequence 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.

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. The pace of a section per week allows time to assign and grade a significant number of problems, since in many ways this is a writing class. In practice, we usually begin the second semester with group homomorphisms and factor groups, and end with geometric constructions.

Chapters 7, 8, and 10 provide the basis for a third semester’s work on Galois theory, which we do at the graduate level. The fourth edition has added a chapter and some harder problems to fill out this material.

FEAtures

*Progresses students from writing proofs in the familiar setting of the integers to dealing with abstract concepts once they have gained some confidence. Separating the two hurdles of learning to write proofs and grasping abstract concepts makes the subject more accessible.

*Makes a concerted effort throughout to develop key examples in detail before introducing the relevant abstract definitions. For example, cyclic groups are introduced in Chapter 1 in the context of number theory, and permutations are studied in Chapter 2, before abstract groups are introduced in Chapter 3. The ring of integers and rings of polynomials are covered before the general notion of a ring is introduced in Chapter 5.

*Provides chapter introductions and notes that give motivation and historical context while tying the subject matter in with the broader picture. The text emphasizes the historical connections to the solution of polynomial equations and to the theory of numbers. For strong classes, there is a complete treatment of Galois theory, and for honors students, there are optional sections on advanced number theory topics.

*Recognizes the developing maturity of students by raising the writing level as the book progresses. The first two chapters (on the integers and functions) contain full details, in addition to comments on techniques of proof. The intermediate chapters on groups, rings, and fields are written at a standard undergraduate level. The final four chapters (on the structure of groups, Galois theory, and unique factorization) are written at a more demanding level, consistent with material usually considered to be at a graduate/undergraduate level.

*Includes such optional topics as finite fields, the Sylow theorems, finite abelian groups, the simplicity of PSL2(F), the classification of groups of small order via semidirect products, Euclidean domains, unique factorization domains, cyclotomic polynomials, arithmetic functions, Moebius inversion, quadratic reciprocity, primitive roots, and diophantine equations.

*Offers an extensive set of exercises that provides ample opportunity for students to develop their ability to write proofs. A (free) study guide is available online, with complete solutions to numerous supplementary problems.S

FROM THE PREFACES

In the fourth edition we have added material to emphasize one of the key features of the book: the rising level of expectations as the students learn the subject. In the early chapters, we have added a few examples and exercises; in Chapters 7 and 8 we have added a significant number of more difficult exercises. We have also added new material on groups in Chapter 10.

The third 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 measurably improved the book.

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. 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.