Introduction to Abstract Algebra

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ISBN 9781420063714
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  • Looks at abstract algebra as the main tool underlying discrete mathematics and the digital world
  • Uses semigroups and monoids as stepping stones to present the concepts of groups and rings
  • Presents the fundamentals of abstract algebra, before offering deeper coverage of group and ring theory
  • Provides examples of abstract algebra concepts in matrices and calculus
  • Contains numerous exercises of varying levels of difficulty, chapter notes that point out variations in notation and approach, and study projects that cover an array of applications and developments of the theory
  • Includes a solutions manual for qualifying instructors
  • Summary

    Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles.

    A Quick Introduction to Algebra

    The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level.

    The Basics of Abstract Algebra for a First-Semester Course

    Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange’s theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley’s theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields.

    Advanced Topics for a Second-Semester Course

    The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups.

    Table of Contents

    Ordering Numbers
    The Well-Ordering Principle
    The Division Algorithm
    Greatest Common Divisors
    The Euclidean Algorithm
    Primes and Irreducibles
    The Fundamental Theorem of Arithmetic
    Specifying Functions
    Composite Functions
    Linear Functions
    Semigroups of Functions
    Injectivity and Surjectivity
    Groups of Permutations
    Kernel and Equivalence Relations
    Equivalence Classes
    Rational Numbers
    The First Isomorphism Theorem for Sets
    Modular Arithmetic
    Groups and Monoids
    Componentwise Structure
    Submonoids and Subgroups
    Multiplication Tables
    Normal Subgroups
    The First Isomorphism Theorem for Groups
    The Law of Exponents
    Cayley’s Theorem
    Ring Homomorphisms
    Quotient Rings
    Polynomial Rings
    Integral Domains
    Polynomials over Fields
    Principal Ideal Domains
    Irreducible Polynomials
    Lagrange Interpolation
    Fields of Fractions
    Factorization in Integral Domains
    Noetherian Domains
    Unique Factorization Domains
    Roots of Polynomials
    Splitting Fields
    Uniqueness of Splitting Fields
    Structure of Finite Fields
    Galois Fields
    Representing a Ring
    Direct Sums
    Free Modules
    Vector Spaces
    Abelian Groups
    Group Actions
    Transitive Actions
    Fixed Points
    Faithful Actions
    Alternating Groups
    Sylow Theorems
    Latin Squares
    Quasigroup Homomorphisms
    Quasigroup Homotopies
    Principal Isotopy
    Exercises, Study Projects, and Notes appear at the end of each chapter.

    Editorial Reviews

    … The author goes the extra mile to build algebraic concepts by confronting the pedagogic and logical sequence groups-first or rings-first dilemma … a perfect pure math precursor to Grillet and Knapp’s works. … The book’s well-thought out sequence supports a set of useful statements on how to use its 11 chapters in a course … The book is also outstanding for self-study. … I recommend this book as second to none on abstract algebra for its content, style, and expository efficiency.
    Computing Reviews, January 2011

    … a careful treatment of the principal topics of abstract algebra … This is an attractive book which could be read by everybody because the author supposes not so much knowledge from the reader and gives all the necessary information to continue the reading from [one] chapter to the next. The approach used by the author to introduce modules and group actions is new and innovative. The book is well written … students and even experienced researchers may benefit strongly from this book. …
    —IACR Book Reviews, October 2010

    … This compact book covers topics one would expect to find in an abstract algebra text. … Smith’s approach is carefully implemented, and topics flow logically from one chapter to the next. The writing is careful and rigorous, yet accessible to hardworking students. The problems are collected at the end of each chapter in two sets, with one set made up of shorter exercises. … This is an ideal text for an abstract algebra course comprised of mathematics students or CS students who have either a strong minor or second major in mathematics. …
    Computing Reviews, December 2009

    One can trace the author’s research interests to the border between algebra and category theory, which gives the textbook its unique flavour.
    EMS Newsletter, March 2009

    The book is well written and flows well. Readers looking for an alternative approach to abstract algebra should consider this volume.
    —J.R. Burke, Gonzaga University, CHOICE, July 2009, Vol. 46, No. 11

    This book is well written, interesting to read, and the proofs and examples are clear and clean.
    —David F. Anderson, Mathematical Reviews, 2009e