Contemporary Abstract Algebra, 8th Edition by Joseph Gallian is a comprehensive textbook on the subject of abstract algebra. It is designed for students studying mathematics at the undergraduate level and aims to provide a thorough understanding of the fundamental concepts of abstract algebra.

The 8th edition of the textbook has been updated to include the latest developments in the field and to incorporate a more streamlined presentation of the material. The book is divided into nine chapters that cover a wide range of topics, including groups, rings, fields, and Galois theory.

**Table of Contents**

** PART 1 Integers and Equivalence Relations 1**

- Properties of Integers
- Modular Arithmetic
- Complex Numbers
- Mathematical Induction
- Equivalence Relations
- Functions (Mappings)
- Exercises

**PART 2 **

**Introduction to Groups **

- Symmetries of a Square
- The Dihedral Groups
- Exercises
- Biography of Niels Abel

**Groups **

- Definition and Examples of Groups
- Elementary
- Properties of Groups
- Historical Note
- Exercises

**Finite Groups; Subgroups**

- Terminology and Notation
- Subgroup Tests
- Examples of Subgroups
- Exercises

.

**Cyclic Groups**

- Classification of Subgroups of Cyclic Groups
- Exercises
- Biography of James Joseph Sylvester
- Supplementary Exercises for Chapters 1–4

**Permutation Groups **

- Definition and Notation
- Cycle Notation
- Properties of Permutations
- A Check-Digit Scheme Based on D5
- Exercises
- Biography of Augustin Cauchy

**Isomorphisms **

- Motivation
- Definition and Examples
- Cayley’s Theorem
- Properties of Isomorphisms
- Automorphisms
- Exercises
- Biography of Arthur Cayley

**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
- Biography of Joseph Lagrange

**External Direct Products **

- Definition and Examples
- Properties of External Direct
- The Group of Units Modulo n as an External Direct
- Applications
- Exercises
- Biography of Leonard Adleman
- Supplementary Exercises for Chapters 5–8

**Normal Subgroups and Factor Groups **

- Normal Subgroups
- Factor Groups
- Applications of Factor Groups
- Internal Direct Products
- Exercises
- Biography of Évariste Galois

**Group Homomorphisms **

- Definition and Examples
- Properties of Homomorphisms
- The First Isomorphism Theorem
- Exercises
- Biography of Camille Jordan

**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 3 Rings **

**Introduction to Rings **

- Motivation and Definition
- Examples of Rings
- Properties of Rings
- Subrings
- Exercises
- Biography of I. N. Herstein

**Integral Domains**

- Definition and Examples of Fields
- Characteristic of a Ring
- Exercises
- Biography of Nathan Jacobson

**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 12–14 281

**Ring Homomorphisms **

- Definition and Examples
- Properties of Ring
- Homomorphisms
- The Field of Quotients
- Exercises 292

**Polynomial Rings **

- Notation and Terminology
- The Division Algorithm and Consequences
- Exercises
- Biography of Saunders Mac Lane

**Factorization of Polynomials **

- Reducibility Tests
- Irreducibility Tests
- Unique
- Factorization in Z[x]
- Weird Dice: An Application of Unique
- Factorization
- Exercises

- Biography of Serge Lang

**Divisibility in Integral Domains**

- Irreducibles, Primes 328 |
- 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 4 Fields **

**Vector Spaces **

- Definition and Examples
- Subspaces
- Linear Independence
- Exercises
- Biography of Emil Artin
- Biography of Olga Taussky-Todd

**Extension Fields **

- The Fundamental Theorem of Field Theory
- Splitting Fields
- Zeros of an Irreducible Polynomial
- Exercises
- Biography of Leopold Kronecker

**Algebraic Extensions **

- Characterization of Extensions
- Finite Extensions
- Properties of Algebraic Extensions
- Exercises
- Biography of Irving Kaplansky

**Finite Fields **

- Classification of Finite Fields
- Structure of Finite Fields
- Subfields of a Finite Field
- Exercises
- Biography of L. E. Dickson

**Geometric Constructions **

- Historical Discussion of Geometric Constructions
- Constructible Numbers
- Angle-Trisectors and Circle-Squarers
- Exercises
- Supplementary Exercises for Chapters 19–23

**PART 5 Special Topics **

**Sylow Theorems**

- Conjugacy Classes
- The Class Equation
- The Probability That Two Elements Commute
- The Sylow Theorems
- Applications of Sylow Theorems
- Exercises
- Biography of Ludwig Sylow

**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

**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.

**Symmetry Groups **

- Isometries
- Classification of Finite Plane Symmetry Groups
- Classification of Finite Groups of Rotations in R3
- Exercises

**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

**Symmetry and Counting **

- Motivation
- Burnside’s Theorem
- Applications
- Group Action
- Exercises
- Biography of William Burnside

**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 Erdó´s

**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

**An Introduction to Galois Theory **

- Fundamental Theorem of Galois Theory
- Solvability of Polynomials by Radicals
- Insolvability of a Quintic
- Exercises
- Biography of Philip Hall

**Cyclotomic Extensions **

- Motivation
- Cyclotomic Polynomials
- The Constructible Regular n-gons
- Exercises

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Contemporary Abstract Algebra by Joseph Gallian 8th Edition provides an introduction to the subject of abstract algebra and its various applications. The chapter provides a brief overview of the history of abstract algebra, its evolution from the study of numbers to the study of structures, and its current status as a key area of mathematical research.

It is dedicated to the study of groups, one of the most important concepts in abstract algebra. The chapter covers the definition of a group, subgroups, normal subgroups, cosets, Lagrange’s Theorem, and the isomorphism theorems. It also includes a discussion of cyclic groups, permutation groups, and the Sylow theorems.

This Book is focused on rings and includes a discussion of the definition of a ring, integral domains, fields, and unique factorization domains. The chapter also covers ideals, quotient rings, and polynomial rings.

It covers fields and Galois’s theory, including the definition of a field and its various properties. The chapter provides a comprehensive treatment of Galois theory, including Galois extensions, the Galois correspondence, and the solvability of polynomials by radicals.

Contemporary Abstract Algebra by Joseph Gallian 8th Edition is dedicated to the study of modules and covers the definition of a module, submodules, direct sums, and the structure theorem for modules. The chapter also includes a discussion of the primary decomposition theorem and the Jordan-Hölder theorem.

This focuses on the study of linear algebra, including the definition of a vector space, subspaces, bases, and dimensions. The chapter covers linear transformations, determinants, eigenvalues, and eigenvectors.

This Book is dedicated to the study of inner product spaces, including the definition of an inner product space, orthogonality, orthonormal bases, and the Gram-Schmidt process. The chapter also includes a discussion of the spectral theorem for normal matrices.

Contemporary Algebra Textbook covers the theory of fields and its applications to coding theory and cryptography. The chapter includes a discussion of error-correcting codes, cyclic codes, and the RSA cryptosystem.

Finally, It provides a summary of the key concepts covered in the book and includes a number of review exercises to help students test their understanding of the material.

In conclusion, Contemporary Abstract Algebra, 8th Edition by Joseph Gallian is a comprehensive and well-written textbook that provides a thorough understanding of the fundamental concepts of abstract algebra. The book is designed for students at the undergraduate level and is an excellent resource for anyone seeking to deepen their knowledge of this important area of mathematics.