Through this book, upper undergraduate mathematics majors will master a challenging yet rewarding subject, and approach advanced studies in algebra, number theory and geometry with confidence. Groups, rings and fields are covered in depth with a strong emphasis on irreducible polynomials, a fresh approach to modules and linear algebra, a fresh take on Gröbner theory, and a group theoretic treatment of Rejewski's deciphering of the Enigma machine. It includes a detailed treatment of the basics on finite groups, including Sylow theory and the structure of finite abelian groups. Galois theory and its applications to polynomial equations and geometric constructions are treated in depth. Those interested in computations will appreciate the novel treatment of division algorithms. This rigorous text 'gets to the point', focusing on concisely demonstrating the concept at hand, taking a 'definitions first, examples next' approach. Exercises reinforce the main ideas of the text and encourage students' creativity.
- Balances accessibility with rigor, allowing motivated students to gain mastery of advanced topics
- Takes a 'definitions first, examples next' approach, striking the right balance between abstract material and its motivation
- Prepares students for advanced studies in mathematics by offering a uniquely wide base of knowledge in key topics
- Coverage includes: finite abelian groups, Sylow theory, semi-direct products, solvable groups, unique factorization, irreducibility of polynomials, Galois theory and solvability by radicals, ruler and compass constructions, module theory over principal ideal domains, and a novel approach to Gröbner bases
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Reviews & endorsements
'This is a very good book, which provides an excellent introduction to modern algebra for senior undergraduate or beginning graduate students. The book includes a thorough coverage of the standard topics in the theories of groups, rings, fields, modules and Galois theory, taking a conceptual approach to algebra. For instance, the group theory part focuses on group actions, the ring theory exposition very appropriately stresses unique factorization properties, and the Galois theory part details some rather conceptual applications. Some of the less standard, very interesting topics are also present, including the breaking of the Enigma machine, as well as an in-depth look at division algorithms, including Gröbner bases. The book includes numerous exercises. All in all, a great new algebra text!' Lenny Fukshansky, Claremont McKenna College
'An excellent textbook for an advanced undergraduate or a beginning graduate course on abstract algebra. Includes a lucid discussion of all core topics in group theory, commutative ring theory, Galois theory, and modules over principal ideal domains. I would describe this book as a simplified version of the classical textbook by Dummit and Foote.' Mihran Papikian, Pennsylvania State University
'The 'comprehensive' in the title is no joke: this book walks the reader through a sea of detailed examples and computations in abstract algebra. These, and the exercises, are well thought out and will appeal to the student who likes a very hands-on kind of textbook. The group actions and division algorithms chapters are my personal favourites, as the computational nature of those topics plays to the strengths of the authors.' Nick Gurski, Case Western Reserve University
'This is a great introduction to abstract algebra for graduate students and mathematically mature undergraduates.' Thomas Garrity, Williams College
'Lawrence and Zorzitto's treatment of Abstract Algebra is lucid and thorough. I am particularly pleased to see the inclusion of Gröbner basis theory in a way that is accessible to introductory students, as it makes possible the exploration of polynomial ideals to great depth.' Jeffrey Clark, Elon University
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Product details
- Date Published: April 2021
- format: Hardback
- isbn: 9781108836654
- length: 638 pages
- dimensions: 253 x 194 x 33 mm
- weight: 1.52kg
- availability: Available
Table of Contents
Contents
Preface
1. A refresher on the integers
2. A first look at groups
3. Groups acting on sets
4. Basics on rings-mostly commutative
5. Primes and unique factorization
6. Algebraic field extensions
7. Applications of galois theory
8. Modules over principal ideal domains
9. Division algorithms
Appendix A: Infinite sets.
Look Inside
Authors
John W. Lawrence, University of Waterloo, Ontario
John W. Lawrence is Professor Emeritus at the University of Waterloo. He was born in Ottawa, Canada, and received degrees from Carleton University and McGill University. After a year of postdoctoral work at the University of Chicago, he joined the Pure Mathematics Department of the University of Waterloo. He now lives with his wife Louise, in Thornhill Canada, where he continues his research in mathematics and probability.
Frank A. Zorzitto, University of Waterloo, Ontario
Frank A. Zorzitto is Professor Emeritus at the University of Waterloo. After receiving his doctorate at Queen's University in Kingston, Canada, he served for forty years as a researcher in algebra and a professor in the Pure Mathematics Department at the University of Waterloo, and served as Department Chair for twelve years. In recognition of his commitment to the education of students, he received the University's Distinguished Teaching Award. Upon his retirement he was made an Honorary Member of the University. Currently he offers an online course to high school math teachers, based on his e-book A Taste of Number Theory. He continues to teach and to write in his retirement.