Abstract Algebra

This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven. The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader’s appetite for further study. This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.

Table of Contents

Chapter 1. A Potpourri of Preliminary Topics

Chapter 2. Groups — Part 1

Chapter 3. Rings — Part 1

Chapter 4. Vector Spaces — Part 1

Chapter 5. Fields — Part 1

Chapter 6. Groups — Part 2

Chapter 7. Rings — Part 2

Chapter 8. Fields — Part 2

Chapter 9. Galois Theory: Fields+Groups

Chapter 10. Vector Spaces — Part 2

Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces

Chapter 12. Groups — Part 3

Chapter 13. Modules — Part 2: Multilinear Algebra

Chapter 14. Additional Topics in Brief

Review

It will come as no surprise that the material is presented in a clear and flawless manner; in addition, there are many exercises and an extensive index. –Franz Lemmermeyer, zbMATH Open

A quick review of these archives alone will show that textbooks for undergraduate abstract algebra courses are not in short supply. Several of them are excellent, and as an instructor, I have an embarrassment of riches in choosing for my course. I expect this text will be on that list when next I get to teach the subject. Silverman’s dedication says, “This one is for the next generation.” Indeed, this is a wonderful resource for training the next generation of mathematicians. –Michele Intermont, Kalamazoo College

About the Author

Joseph H. Silverman, Brown University, Providence, RI