Introduction to Abstract Algebra

Jonathan D. H. Smith

August 20, 2008 by Chapman and Hall/CRC
Textbook - 344 Pages - 43 B/W Illustrations
ISBN 9781420063714 - CAT# C0637
Series: Textbooks in Mathematics

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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.