Introduction to Abstract Algebra.
Introduction to Abstract Algebra
Jonathan D. H. Smith
6000 NW Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487
9781498731614, $89.95, 352pp, www.amazon.com
Synopsis: Now in a newly updated and expanded second edition, "Introduction to Abstract Algebra" by academician Jonathan D. H. Smith (Iowa State University, Ames, Iowa) presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It avoids the usual groups first/rings first dilemma by introducing semigroups and monoids, the multiplicative structures of rings, along with groups. This new edition of "Introduction to Abstract Algebra" covers applications from biology, science, and engineering. It offers numerous updates based on feedback from first edition adopters, as well as improved and simplified proofs of a number of important theorems. Many new exercises have been added, while new study projects examine skewfields, quaternions, and octonions.
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. These three chapters provide a quick introduction to algebra, sufficient to exhibit irrational numbers or to gain a taste of cryptography. Chapters four through seven cover abstract groups and monoids, orthogonal groups, stochastic matrices, Lagrange's theorem, groups of units of monoids, homomorphisms, rings, and integral domains. The first seven chapters provide basic coverage of abstract algebra, suitable for a one-semester or two-quarter course.
Each chapter includes 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. The final chapters deal with slightly more advanced topics, suitable for a second-semester or third-quarter course. These chapters delve deeper into the theory of rings, fields, and groups. They discuss modules, including vector spaces and abelian groups, group theory, and quasigroups.
As a curriculum textbook "Introduction to Abstract Algebra" is suitable for use in an undergraduate course on abstract algebra for mathematics, computer science, and education majors, along with students from other STEM fields.
Critique: A complete course of instruction under one cover, "Introduction to Abstract Algebra" is a standard text that should be a part of every community and academic library Mathematics reference collection in general, and Algebraic Studies supplemental reading in particular. For students and non-specialist general readers with an interest in higher mathematics, it should be noted that "Introduction to Abstract Algebra" is also available in a Kindle edition ($71.99).