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.
The Well-Ordering Principle
The Division Algorithm
Greatest Common Divisors
The Euclidean Algorithm
Primes and Irreducibles
The Fundamental Theorem of Arithmetic
Semigroups of Functions
Injectivity and Surjectivity
Groups of Permutations
Kernel and Equivalence Relations
The First Isomorphism Theorem for Sets
Groups and Monoids
Submonoids and Subgroups
The First Isomorphism Theorem for Groups
The Law of Exponents
Polynomials over Fields
Principal Ideal Domains
Fields of Fractions
Factorization in Integral Domains
Unique Factorization Domains
Roots of Polynomials
Uniqueness of Splitting Fields
Structure of Finite Fields
Representing a Ring
Exercises, Study Projects, and Notes appear at the end of each chapter.
… 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
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