You are previewing Introduction to Abstract Algebra, 4th Edition.
O'Reilly logo
Introduction to Abstract Algebra, 4th Edition

Book Description

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:

  • The treatment of nilpotent groups, including the Frattini and Fitting subgroups

  • Symmetric polynomials

  • The proof of the fundamental theorem of algebra using symmetric polynomials

  • The proof of Wedderburn's theorem on finite division rings

  • The proof of the Wedderburn-Artin theorem

  • Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.

    Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

    Table of Contents

    1. Cover
    2. Title Page
    3. Copyright
    4. Preface
    5. Acknowledgments
    6. Notation Used in the Text
    7. A Sketch of the History of Algebra to 1929
    8. Chapter 0: Preliminaries
      1. 0.1 Proofs
      2. 0.2 Sets
      3. 0.3 Mappings
      4. 0.4 Equivalences
    9. Chapter 1: Integers and Permutations
      1. 1.1 Induction
      2. 1.2 Divisors and Prime Factorization
      3. 1.3 Integers Modulo n
      4. 1.4 Permutations
      5. 1.5 An Application to Cryptography
    10. Chapter 2: Groups
      1. 2.1 Binary Operations
      2. 2.2 Groups
      3. 2.3 Subgroups
      4. 2.4 Cyclic Groups and the Order of an Element
      5. 2.5 Homomorphisms and Isomorphisms
      6. 2.6 Cosets and Lagrange's Theorem
      7. 2.7 Groups of Motions and Symmetries
      8. 2.8 Normal Subgroups
      9. 2.9 Factor Groups
      10. 2.10 The Isomorphism Theorem
      11. 2.11 An Application to Binary Linear Codes
    11. Chapter 3: Rings
      1. 3.1 Examples and Basic Properties
      2. 3.2 Integral Domains and Fields
      3. 3.2 Exercises
      4. 3.3 Ideals and Factor Rings
      5. 3.4 Homomorphisms
      6. 3.5 Ordered Integral Domains
    12. Chapter 4: Polynomials
      1. 4.1 Polynomials
      2. 4.2 Factorization of Polynomials over a Field
      3. 4.3 Factor Rings of Polynomials over a Field
      4. 4.4 Partial Fractions
      5. 4.5 Symmetric Polynomials
      6. 4.6 Formal Construction of Polynomials
    13. Chapter 5: Factorization in Integral Domains
      1. 5.1 Irreducibles and Unique Factorization
      2. 5.2 Principal Ideal Domains
    14. Chapter 6: Fields
      1. 6.1 Vector Spaces
      2. 6.2 Algebraic Extensions
      3. 6.3 Splitting Fields
      4. 6.4 Finite Fields
      5. 6.5 Geometric Constructions
      6. 6.6 The Fundamental Theorem of Algebra
      7. 6.7 An Application to Cyclic and BCH Codes
    15. Chapter 7: Modules over Principal Ideal Domains
      1. 7.1 Modules
      2. 7.2 Modules Over a PID
    16. Chapter 8: p-Groups and the Sylow Theorems
      1. 8.1 Products and Factors
      2. 8.2 Cauchy's Theorem
      3. 8.3 Group Actions
      4. 8.4 The Sylow Theorems
      5. 8.5 Semidirect Products
      6. 8.6 An Application to Combinatorics
    17. Chapter 9: Series of Subgroups
      1. 9.1 The Jordan–Hölder Theorem
      2. 9.2 Solvable Groups
      3. 9.3 Nilpotent Groups
    18. Chapter 10: Galois Theory
      1. 10.1 Galois Groups and Separability
      2. 10.2 The Main Theorem of Galois Theory
      3. 10.3 Insolvability of Polynomials
      4. 10.4 Cyclotomic Polynomials and Wedderburn's Theorem
    19. Chapter 11: Finiteness Conditions for Rings and Modules
      1. 11.1 Wedderburn's Theorem
      2. 11.2 The Wedderburn–Artin Theorem
    20. Appendices
      1. Appendix A Complex Numbers
      2. Appendix B Matrix Algebra
      3. Appendix C Zorn's Lemma
      4. Appendix D Proof of the Recursion Theorem
    21. Bibliography
    22. Selected Answers
    23. Index