Table of Contents

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

About the Author

Iowa State University, Ames, USA

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