Contemporary Abstract Algebra, Tenth Edition
For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.
The author presents the concepts and methodologies of contemporary abstract algebra as used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop but rarely do. Applications are included to illustrate the utility of the abstract concepts.
Examples and exercises are the heart of the book. Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The exercises often foreshadow definitions, concepts, and theorems to come.
Changes for the tenth edition include new exercises, new examples, new quotes, and a freshening of the discussion portions. The hallmark features of previous editions of the book are enhanced in this edition. These include:
- A good mixture of approximately 1900 computational and theoretical exercises, including computer exercises, that synthesize concepts from multiple chapters
- Approximately 300 worked-out examples from routine computations to the challenging
- Many applications from scientific and computing fields and everyday life
- Historical notes and biographies that spotlight people and events
- Motivational and humorous quotations
- Numerous connections to number theory and geometry
While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor's Solutions Manual written by the author has comprehensive solutions for all exercises and some alternative solutions to develop a critical thought and deeper understanding. It is available from CRC Press only. The Student Solution Manual has comprehensive solutions for all odd-numbered exercises and many even-numbered exercises.
Author
Joseph A. Gallian earned his PhD from Notre Dame. In addition to receiving numerous national awards for his teaching and exposition, he has served terms as the Second Vice President, and 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.
Author(s): Joseph A. Gallian
Edition: 10
Publisher: CRC Press
Year: 2021
Language: English
Pages: 654
Tags: modern algebra, modern, higher math, higher, math, mathematics, maths
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Notations
Preface
0 Preliminaries
Properties of Integers
Modular Arithmetic
Complex Numbers
Mathematical Induction
Equivalence Relations
Functions (Mappings)
Exercises
1 Introduction to Groups
Symmetries of a Square
The Dihedral Groups
Bibliography of Niels 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
Bibliography of James Joseph Sylvester
5 Permutation Groups
Definitions and Notation
Cycle Notation
Properties of Permutations
A Check-Digit Scheme Based on D_5
Bibliography of Augustin Cauchy
Bibliography of Alan Turing
6 Isomorphisms
Motivation
Definition and Examples
Properties of Isomorphisms
Automorphisms
Cayley’s Theorem
Exercises
Bibliography 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
An Application of Cosets to the Rubik's Cube
Exercises
Bibliography 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
Bibliography of Leonard Adleman
9 Normal Subgroups and Factor Groups
Normal Subgroups
Factor Groups
Applications of Factor Groups
Internal Direct Products
Exercises
Bibliography of Évariste Galois
10 Group Homomorphisms
Definition and Examples
Properties of Homomorphisms
The First Isomorphism Theorem
Exercises
Bibliography 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
12 Introduction to Rings
Motivation and Definition
Examples of Rings
Properties of Rings
Subrings
Exercises
Bibliography of I. N. Herstein
13 Integral Domains
Definition and Examples
Fields
Characteristic of a Ring
Exercises
14 Ideals and Factor Rings
Ideals
Factor Rings
Prime Ideals and Maximal Ideals
Exercises
Bibliography of Richard Dedekind
Bibliography of Emmy Noether
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
17 Factorization of Polynomials
Reducibility Tests
Irreducibility Tests
Unique Factorization in Z[x]
Weird Dice: An Application of Unique Factorization
Exercises
Bibliography of Serge Lang
18 Divisibility in Integral Domains
Irreducibles, Primes
Historical Discussion of Fermat's Last Theorem
Unique Factorization Domains
Euclidean Domains
Exercises
Bibliography of Sophie Germain
Bibliography of Andrew Wiles
Bibliography of Pierre de Fermat
19 Extension Fields
The Fundamental Theorem of Field Theory
Splitting Fields
Zeros of an Irreducible Polynomial
Exercises
Bibliography of Leopold Kronecker
20 Algebraic Extensions
Characterization of Extensions
Finite Extensions
Properties of Algebraic Extensions
Exercises
Bibliography of Ernst Steinitz
21 Finite Fields
Classification of Finite Fields
Structure of Finite Fields
Subfields of a Finite Field
Exercises
Bibliography of L. E. Dickson
Bibliography of E. H. Moore
22 Geometric Constructions
Historical Discussion of Geometric Constructions
Constructible Numbers
Angle-Trisectors and Circle-Squarers
Exercises
23 Sylow Theorems
Conjugacy Classes
The Class Equation
The Sylow Theorems
Applications of Sylow Theorems
Exercises
Bibliography of Ludwig Sylow
24 Finite Simple Groups
Historical Background
Nonsimplicity Tests
The Simplicity of A_5
The Fields Medal
The Cole Prize
Exercises
Bibliography of Michael Aschbacher
Bibliography of Daniel Gorenstein
Bibliography of John Thompson
25 Generators and Relations
Motivation
Definitions and Notation
Free Group
Generators and Relations
Classification of Groups of Order Up to 15
Characterization of Dihedral Groups
Exercises
Bibliography of Marshall Hall, Jr.
26 Symmetry Groups
Isometries
Classification of Finite Plane Symmetry Groups
Classification of Finite Groups of Rotations in R^3
Exercises
27 Symmetry and Counting
Motivation
Burnside's Theorem
Applications
Group Action
Exercises
Bibliography of William Burnside
28 Cayley Digraphs of Groups
Motivation
The Cayley Digraph of a Group
Hamiltonian Circuits and Paths
Some Applications
Exercises
Bibliography of William Rowan Hamilton
Bibliography of Paul Erdős
29 Introduction to Algebraic Coding Theory
Motivation
Linear Codes
Parity-Check Matrix Decoding
Coset Decoding
Historical Note
Exercises
Bibliography of Richard W. Hamming
Bibliography of Jessie MacWilliams
Bibliography of Vera Pless
30 An Introduction to Galois Theory
Fundamental Theorem of Galois Theory
Solvability of Polynomials by Radicals
Insolvability of a Quintic
Exercises
31 Cyclotomic Extensions
Motivation
Cyclotomic Polynomials
The Constructible Regular n-gons
Exercises
Bibliography of Carl Friedrich Gauss
Bibliography of Manjul Bhargava
Selected Answers
Index
