Professional

# Contemporary Abstract Algebra, 8th Edition

### Overview

CONTEMPORARY ABSTRACT ALGEBRA, EIGHTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.

### Features and Benefits

• "Why Is Abstract Algebra Useful?" essay on the author's website shows students the importance of studying abstract algebra and its applicability to everyday life. This motivational essay can be incorporated into a lesson on the first day of class, or can serve as a motivational tool further along in the course.
• Coverage of proof writing is provided on the instructor's website in the updated essay "Advice for students learning proofs." This essay provides basic guidance for students encountering proof writing for the first time or needing to freshen their skills.
• This text focuses on abstract algebra as a contemporary subject, with concepts and methodologies currently being used by computer scientists, physics, and chemists.
• Extensive coverage of groups, rings, and fields, plus a variety of non-traditional special topics.
• A good mixture of now more that 1750 computational and theoretical exercises appearing in each chapter and in Supplementary Exercise sets that synthesize concepts from multiple chapters.
• Emphasis on computation and proof writing, with an abundance of exercises to help students develop both skills.
• Lines from popular songs, poems, and quotations give the text a fresh, contemporary feel and keep students engaged.

PART I: INTEGERS AND EQUIVALENCE RELATIONS.
Preliminaries. Properties of Integers. Complex Numbers. Modular Arithmetic. Mathematical Induction. Equivalence Relations. Functions (Mappings). Exercises.
PART I: GROUPS.
1. Introduction to Groups.
Symmetries of a Square. The Dihedral Groups. Exercises. Biography of Neils Abel
2. Groups.
Definition and Examples of Groups. Elementary Properties of Groups. Historical Note. Exercises.
3. Finite Groups; Subgroups.
Terminology and Notation. Subgroup Tests. Examples of Subgroups. Exercises.
4. Cyclic Groups.
Properties of Cyclic Groups. Classification of Subgroups of Cyclic Groups. Exercises. Biography of J. J. Sylvester. Supplementary Exercises for Chapters 1-4.
5. Permutation Groups.
Definition and Notation. Cycle Notation. Properties of Permutations. A Check-Digit Scheme Based on D5. Exercises. Biography of Augustin Cauchy.
6. Isomorphisms.
Motivation. Definition and Examples. Cayley''s Theorem. Properties of Isomorphisms.
Automorphisms. Exercises. Biography of Arthur Cayley.
7. Cosets and Lagrange''s Theorem.
Properties of Cosets. Lagrange''s Theorem and Consequences. An Application of Cosets to Permutation Groups. The Rotation Group of a Cube and a Soccer Ball. Exercises. Biography of Joseph Lagrange.
8. External Direct Products.
Definition and Examples. Properties of External Direct Products. The Group of Units Modulo n as an External Direct Product. Applications. Exercises.
Biography of Leonard Adleman. Supplementary Exercises for Chapters 5-8
9. Normal Subgroups and Factor Groups.
Normal Subgroups. Factor Groups. Applications of Factor Groups. Internal Direct Products. Exercises. Biography of Évariste Galois
10. Group Homomorphisms.
Definition and Examples. Properties of Homomorphisms. The First Isomorphism Theorem. Exercises. Biography of Camille Jordan.
11. Fundamental Theorem of Finite Abelian Groups.
The Fundamental Theorem. The Isomorphism Classes of Abelian Groups. Proof of the Fundamental Theorem. Exercises. Supplementary Exercises for Chapters 9-11.
PART III: RINGS.
12. Introduction to Rings.
Motivation and Definition. Examples of Rings. Properties of Rings. Subrings. Exercises. Biography of I. N. Herstein.
13. Integral Domains.
Definition and Examples. Fields. Characteristic of a Ring. Exercises. Biography of Nathan Jacobson.
14. Ideals and Factor Rings.
Ideals. Factor Rings. Prime Ideals and Maximal Ideals. Exercises.
Biography of Richard Dedekind. Biography of Emmy Noether. Supplementary Exercises for Chapters 12-14.
15. Ring Homomorphisms.
Definition and Examples. Properties of Ring Homomorphisms. The Field of Quotients.
Exercises.
16. Polynomial Rings.
Notation and Terminology. The Division Algorithm and Consequences. Exercises.
Biography of Saunders Mac Lane.
17. Factorization of Polynomials.
Reducibility Tests. Irreducibility Tests. Unique Factorization in Z[x]. Weird Dice: An Application of Unique Factorization. Exercises. Biography of Serge Lang.
18. Divisibility in Integral Domains.
Irreducibles, Primes. Historical Discussion of Fermat''s Last Theorem. Unique Factorization Domains. Euclidean Domains. Exercises.
Biography of Sophie Germain. Biography of Andrew Wiles. Supplementary Exercises for Chapters 15-18.
PART IV: FIELDS.
19. Vector Spaces.
Definition and Examples. Subspaces. Linear Independence. Exercises. Biography of Emil Artin. Biography of Olga Taussky-Todd.
20. Extension Fields.
The Fundamental Theorem of Field Theory. Splitting Fields. Zeros of an Irreducible Polynomial. Exercises. Biography of Leopold Kronecker.
21. Algebraic Extensions.
Characterization of Extensions. Finite Extensions. Properties of Algebraic Extensions
Exercises. Biography of Irving Kaplansky.
22. Finite Fields.
Classification of Finite Fields. Structure of Finite Fields. Subfields of a Finite Field.
Exercises. Biography of L. E. Dickson.
23. Geometric Constructions.
Historical Discussion of Geometric Constructions. Constructible Numbers. Angle-Trisectors and Circle-Squarers. Exercises. Supplementary Exercises for Chapters 19-23.
PART V: SPECIAL TOPICS.
24. Sylow Theorems.
Conjugacy Classes. The Class Equation. The Probability That Two Elements Commute. The Sylow Theorems. Applications of Sylow Theorems. Exercises. Biography of Ludvig Sylow.
25. Finite Simple Groups.
Historical Background. Nonsimplicity Tests. The Simplicity of A5. The Fields Medal. The Cole Prize. Exercises. Biography of Michael Aschbacher. Biography of Daniel Gorenstein. Biography of John Thompson.
26. Generators and Relations.
Motivation. Definitions and Notation. Free Group. Generators and Relations. Classification of Groups of Order up to 15. Characterization of Dihedral Groups. Realizing the Dihedral Groups with Mirrors. Exercises. Biography of Marshall Hall, Jr..
27. Symmetry Groups.
Isometries. Classification of Finite Plane Symmetry Groups. Classification of Finite Group of Rotations in R³. Exercises.
28. Frieze Groups and Crystallographic Groups.
The Frieze Groups. The Crystallographic Groups. Identification of Plane Periodic Patterns. Exercises. Biography of M. C. Escher. Biography of George Pólya. Biography of John H. Conway.
29. Symmetry and Counting.
Motivation. Burnside''s Theorem. Applications. Group Action. Exercises. Biography of William Burnside.
30. Cayley Digraphs of Groups.
Motivation. The Cayley Digraph of a Group. Hamiltonian Circuits and Paths. Some Applications. Exercises. Biography of William-Rowan Hamilton. Biography of Paul Erdös.
31. Introduction to Algebraic Coding Theory.
Motivation. Linear Codes. Parity-Check Matrix Decoding. Coset Decoding.
Historical Note: The Ubiquitous Reed-Solomon Codes. Exercises. Biography of Richard W. Hamming. Biography of Jessie MacWilliams. Biography of Vera Pless. 32. An Introduction to Galois Theory. Fundamental Theorem of Galois Theory. Solvability of Polynomials by. Radicals. Insolvability of a Quintic. Exercises. Biography of Philip Hall.
33. Cyclotomic Extensions.
Motivation. Cyclotomic Polynomials. The Constructible Regular n-gons. Exercises. Biography of Carl Friedrich Gauss. Biography of Manjul Bhargava.
Supplementary Exercises for Chapters 24-33.

### What's New

• Over 200 new exercises!
• New Examples!
• Refreshed quotations, historical notes, and biographies!

## Supplements

All supplements have been updated in coordination with the main title. Select the main title's "About" tab, then select "What's New" for updates specific to title's edition.

### Instructor Supplements

#### Solutions Builder  (ISBN-10: 1133609600 | ISBN-13: 9781133609605)

This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. www.cengage.com/solutionbuilder

#### Student Solutions Manual  (ISBN-10: 1133608531 | ISBN-13: 9781133608530)

Prepare for exams and succeed in your mathematics course with this comprehensive solutions manual! Featuring worked out-solutions to the problems in CONTEMPORARY ABSTRACT ALGEBRA, 8th Edition, this manual shows you how to approach and solve problems using the same step-by-step explanations found in your textbook examples.

List Price = \$96.95  | CengageBrain Price = \$96.95  | College Bookstore Wholesale Price = \$72.75

#### Online GAP Manual  (ISBN-10: 1133610404 | ISBN-13: 9781133610403)

Contains an online laboratory manual for the computer algebra system GAP, with exercises tied to the book.

#### Introduction to Advanced Mathematics: A Guide to Understanding Proofs  (ISBN-10: 0547165382 | ISBN-13: 9780547165387)

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed.

List Price = \$52.95  | CengageBrain Price = \$52.95  | College Bookstore Wholesale Price = \$39.75

### Student Supplements

#### Student Solutions Manual  (ISBN-10: 1133608531 | ISBN-13: 9781133608530)

Prepare for exams and succeed in your mathematics course with this comprehensive solutions manual! Featuring worked out-solutions to the problems in CONTEMPORARY ABSTRACT ALGEBRA, 8th Edition, this manual shows you how to approach and solve problems using the same step-by-step explanations found in your textbook examples.

List Price = \$96.95  | CengageBrain Price = \$96.95  | College Bookstore Wholesale Price = \$72.75

#### Online GAP Manual  (ISBN-10: 1133610404 | ISBN-13: 9781133610403)

Contains an online laboratory manual for the computer algebra system GAP, with exercises tied to the book.

List Price = \$.00  | College Bookstore Wholesale Price = \$.00

#### Introduction to Advanced Mathematics: A Guide to Understanding Proofs  (ISBN-10: 0547165382 | ISBN-13: 9780547165387)

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed.

List Price = \$52.95  | CengageBrain Price = \$52.95  | College Bookstore Wholesale Price = \$39.75

## Meet the Author

### Joseph A. Gallian

Joseph Gallian earned his PhD from Notre Dame. In addition to receiving numerous awards for his teaching and exposition, he served, first, as the Second Vice President, and, then, as the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.