Contemporary Abstract Algebra

Höfundur: Joseph Gallian (Útgáfa: 11)
Contemporary Abstract Algebra

Kaup valmöguleikar

Contemporary Abstract Algebra, Eleventh Edition is intended for a course whose main purpose is to enable students to do computations and write proofs. This text stresses the importance of obtaining a solid introduction to the traditional topics, while at the same time presenting abstract algebra as a contemporary and very much active subject, which is currently being used by working physicists, chemists, and computer scientists.

For nearly four 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. Students will learn how to do computations and 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.

Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. 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 appearing in each chapter that synthesizes concepts from multiple chapters Back-of-the-book skeleton solutions and hints to odd-numbered exercises Over 300 worked-out examples ranging from routine computations to the more challenging Links to interactive True/False questions with comments Links to computer exercises that utilize interactive software available on the author's website, stressing guessing and making conjectures Many applications from scientific and computing fields, as well as some from everyday life Numerous historical notes and biographies that spotlight the people and events behind the mathematics Motivational and humorous quotations Hundreds of figures, photographs, and tables Changes to the eleventh edition include new exercises, examples, biographies, and quotes, and an enrichment of the discussion portions.

These changes accentuate and enhance the hallmark features that have made previous editions of the book a comprehensive, lively, and engaging introduction to the subject. While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor’s Solutions Manual offers solutions for all the exercises. A Student's Solution Manual has comprehensive solutions for all odd-numbered exercises, many even-numbered exercises, and numerous alternative solutions as well.

Nánar um bókina

Útgefandi
Taylor & Francis
ISBN
9781040296516
Print ISBN
9781041298250
Format
ePub
Útgáfa
11
Höfundar
Joseph Gallian
Tungumál
English
Útgefið
2025-06-20
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • Cover Page
  • Half-Title Page
  • Series Page
  • Title Page
  • Copyright Page
  • Dedication Page
  • Contents
  • 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
  • Exercises
  • Biography 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
  • Exercises
  • Biography of James Joseph Sylvester
  • 5 Permutation Groups
  • Exercises
  • Biography of Augustin Cauchy
  • Biography of Alan Turing
  • 6 Isomorphisms
  • Motivation
  • Definition and Examples
  • Properties of Isomorphisms
  • Automorphisms
  • Cayley's Theorem
  • 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
  • An Application of Cosets to the Rubik's Cube
  • Exercises
  • Biography of Joseph Lagrange
  • 8 External Direct Products
  • Applications
  • Exercises
  • Biography of Leonard Adleman
  • 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
  • Biography of Georg Frobenius
  • 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 B. L. van der Waerden
  • 14 Ideals and Factor Rings
  • Factor Rings
  • Prime Ideals and Maximal Ideals
  • Exercises
  • Biography of Richard Dedekind
  • Biography of Emmy Noether
  • 15 Ring Homomorphisms
  • Definitions and Examples
  • The Field of Quotients
  • Exercises
  • Biography of Emil Artin
  • 16 Polynomial Rings
  • Notation and Terminology
  • The Division Algorithm and Consequences
  • Exercises
  • Biography of Saunders Mac Lane
  • 17 Factorization of Polynomials
  • 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
  • Biography of Pierre de Fermat
  • 19 Extension Fields
  • The Fundamental Theorem of Field Theory
  • Splitting Fields
  • Zeros of an Irreducible Polynomial
  • Exercises
  • Biography of Leopold Kronecker
  • 20 Algebraic Extensions
  • Characterization of Extensions
  • Finite Extensions
  • Properties of Algebraic Extensions
  • Exercises
  • Biography of Ernst Steinitz
  • 21 Finite Fields
  • Classification of Finite Fields
  • Structure of Finite Fields
  • Subfields of a Finite Field
  • Exercises
  • Biography of L. E. Dickson
  • Biography of E. H. Moore
  • 22 Geometric Constructions
  • Historical Discussion of Geometric Constructions
  • Constructible Numbers
  • Exercises
  • 23 Sylow Theorems
  • Conjugacy Classes
  • The Class Equation
  • The Sylow Theorems
  • Applications of Sylow Theorems
  • Exercises
  • Biography of Ludwig Sylow
  • 24 Finite Simple Groups
  • Historical Background
  • Exercises
  • Biography of Michael Aschbacher
  • Biography of Daniel Gorenstein
  • Biography of John Thompson
  • 25 Generators and Relations
  • Motivation
  • Definitions and Notation
  • Generators and Relations
  • Classification of Groups of Order Up to 15
  • Characterization of Dihedral Groups
  • Exercises
  • Biography of Marshall Hall, Jr.
  • 26 Symmetry Groups
  • Isometries
  • Classification of Finite Plane Symmetry Groups
  • Classification of Finite Groups of Rotations in R3
  • Exercises
  • 27 Symmetry and Counting
  • Motivation
  • Burnside's Theorem
  • Applications
  • Group Action
  • Exercises
  • Biography of William Burnside
  • 28 Cayley Digraphs of Groups
  • Motivation
  • The Cayley Digraph of a Group
  • Exercises
  • Biography of William Rowan Hamilton
  • Biography of Paul Erdős
  • 29 Introduction to Algebraic Coding Theory
  • Motivation
  • Linear Codes
  • Parity-Check Matrix Decoding
  • Coset Decoding
  • Historical Note
  • Exercises
  • Biography of Richard W. Hamming
  • Biography of Jessie MacWilliams
  • Biography of Vera Pless
  • 30 An Introduction to Galois Theory
  • Exercises
  • 31 Cyclotomic Extensions
  • Exercises
  • Biography of Carl Friedrich Gauss
  • Biography of Manjul Bhargava
  • Selected Answers
  • Index