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First Course in Abstract Algebra, A, 8th edition

Published by Pearson (February 21, 2020) © 2021

  • John B Fraleigh University of Rhode Island
  • Neal Brand University of North Texas

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For courses in Abstract Algebra.

A comprehensive approach to abstract algebra

A First Course in Abstract Algebra, 8th Edition covers all the essentials for an in-depth introduction to abstract algebra, and is designed for maximum relevance to future graduate students, future high school teachers and students who intend to work in industry. New co-author Neal Brand has revised this classic text carefully and thoughtfully, drawing on years of experience teaching the course with this text to produce a meaningful and worthwhile update. Students build a firm foundation for more specialized work in algebra through its extensive explanations of the what, how and why behind each method. This revision also includes applied topics such as RSA encryption and coding theory, as well as examples of applying Gröbner bases.

Hallmark features of this title

  • A focus on groups, rings and fields gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.
  • A study of group theory at the start of the text provides an easy transition to axiomatic mathematics.
  • Clear and concise explanations of the theory with well-thought-out examples highlight key points and clarify more difficult concepts.
  • Numerous exercises in almost every section range from routine to very challenging. Exercises are computational, true/false, summarizing proofs, identifying errors, and proving statements.
  • Historical notes written by Victor Katz, an authority on the history of math, provide valuable perspective.

New and updated features of this title

  • UPDATED: Many exercises in the text have been updated and many are new. Most exercise sets are broken down into parts consisting of computations, concepts, and theory.
    • NEW: Answers to parts a), c), e), g), and i) of the 10-part true-false exercises are provided.
  • REVISED: Clarity is enhanced in a few sections, while maintaining the standard users expect from Fraleigh. For example, sections in Part IX on Galois Theory have been rewritten to improve readability.
  • REVISED: Some topics have been reordered to streamline the flow of the book.
  • REVISED: Exercises throughout are refreshed, with somereplaced, some re-worded, and others added.
  • NEW: Applied topics such as RSA encryption and coding theory as well as examples of applying Gröbner bases have been added.

Brief Table of Contents

  • Instructor's Preface
  • Dependence Chart
  • Student's Preface
  1. Sets and Relations

I. GROUPS AND SUBGROUPS

  1. Binary Operations
  2. Groups
  3. Abelian Groups
  4. Nonabelian Examples
  5. Subgroups
  6. Cyclic Groups
  7. Generating Sets and Cayley Digraphs

II. STRUCTURE OF GROUPS

  1. Groups and Permutations
  2. Finitely Generated Abelian Groups
  3. Cosets and the Theorem of Lagrange
  4. Plane Isometries

III. HOMOMORPHISMS AND FACTOR GROUPS

  1. Factor Groups
  2. Factor-Group Computations and Simple Groups
  3. Groups Actions on a Set
  4. Applications of G -Sets to Counting

IV. ADVANCED GROUP THEORY

  1. Isomorphism Theorems
  2. Sylow Theorems
  3. Series of Groups
  4. Free Abelian Groups
  5. Free Groups
  6. Group Presentations

V. RINGS AND FIELDS

  1. Rings and Fields
  2. Integral Domains
  3. Fermat's and Euler's Theorems
  4. Encryption

VI. CONSTRUCTING RINGS AND FIELDS

  1. The Field of Quotients of an Integral Domain
  2. Rings and Polynomials
  3. Factorization of Polynomials over Fields
  4. Algebraic Coding Theory
  5. Homomorphisms and Factor Rings
  6. Prime and Maximal Ideals
  7. Noncommutative Examples

VII. COMMUTATIVE ALGEBRA

  1. Vector Spaces
  2. Unique Factorization Domains
  3. Euclidean Domains
  4. Number Theory
  5. Algebraic Geometry
  6. Gröbner Basis for Ideals

VIII. EXTENSION FIELDS

  1. Introduction to Extension Fields
  2. Algebraic Extensions
  3. Geometric Constructions
  4. Finite Fields

IX. Galois Theory

  1. Introduction to Galois Theory
  2. Splitting Fields
  3. Separable Extensions
  4. Galois Theory
  5. Illustrations of Galois Theory
  6. Cyclotomic Extensions
  7. Insolvability of the Quintic

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