Introduction to Abstract Algebra: (Textbooks in Mathematics)

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:
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

Review :
… 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