2nd Edition

Introduction to Abstract Algebra

By Jonathan D. H. Smith Copyright 2016
    352 Pages 52 B/W Illustrations
    by Chapman & Hall

    Introduction to Abstract Algebra, Second Edition 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 a widely adopted textbook 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.

    This textbook 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.

    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
    Exercises
    Study projects
    Notes

    Functions
    Specifying functions
    Composite functions
    Linear functions
    Semigroups of functions
    Injectivity and surjectivity
    Isomorphisms
    Groups of permutations
    Exercises
    Study projects
    Notes
    Summary

    Equivalence
    Kernel and equivalence relations
    Equivalence classes
    Rational numbers
    The First Isomorphism Theorem for Sets
    Modular arithmetic
    Exercises
    Study projects
    Notes

    Groups and Monoids
    Semigroups
    Monoids
    Groups
    Componentwise structure
    Powers
    Submonoids and subgroups
    Cosets
    Multiplication tables
    Exercises
    Study projects
    Notes

    Homomorphisms
    Homomorphisms
    Normal subgroups
    Quotients
    The First Isomorphism Theorem for Groups
    The Law of Exponents
    Cayley’s Theorem
    Exercises
    Study projects
    Notes

    Rings
    Rings
    Distributivity
    Subrings
    Ring homomorphisms
    Ideals
    Quotient rings
    Polynomial rings
    Substitution
    Exercises
    Study projects
    Notes

    Fields
    Integral domains
    Degrees
    Fields
    Polynomials over fields
    Principal ideal domains
    Irreducible polynomials
    Lagrange interpolation
    Fields of fractions
    Exercises
    Study projects
    Notes

    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
    Exercises
    Study projects
    Notes

    Modules
    Endomorphisms
    Representing a ring
    Modules
    Submodules
    Direct sums
    Free modules
    Vector spaces
    Abelian groups
    Exercises
    Study projects
    Notes

    Group Actions
    Actions
    Orbits
    Transitive actions
    Fixed points
    Faithful actions
    Cores
    Alternating groups
    Sylow Theorems
    Exercises
    Study projects
    Notes

    Quasigroups
    Quasigroups
    Latin squares
    Division
    Quasigroup homomorphisms
    Quasigroup homotopies
    Principal isotopy
    Loops
    Exercises
    Study projects
    Note

    Index

    Biography

    Jonathan Smith is a Professor at Iowa State University. He earned his Ph.D., from Cambridge (England). His research focuses on combinatorics, algebra, and information theory; applications in computer science, physics, and biology.

    "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."
    Reviewer’s Bookwatch, December 2015

     

    Smith’s update to the first edition (CH, Jul'09, 46-6260) is an alternative approach to the usual first semester in higher algebra. The author accomplishes this by including many topics often absent from a first course, such as quasigroups, Noetherian domains, and modules, which, theoretically, are developed alongside their mainstream analogues, like groups, rings, and vector spaces. It is essentially a first semester wink at universal algebra. Smith’s approach to axiomatic systems is few-too-many—he starts with structures with very few axioms, like semigroups and monoids, and continues adding axioms. He finishes with more complex axiomatic systems, like unique factorization domains and fields. The book is very well written and easy to read, flowing naturally from one topic to the next. Numerous supportive homework exercises are also included to help the reader explore further topics. This book will best serve readers with a background in abstract algebra who desire to strengthen their understanding and build bridges between various topics. Unfortunately, because many similar topics are handled in tandem, an inexperienced reader might become confused, especially as many clarifying examples are missing. This book is for readers who want an under the hood view of algebra.
    --A. Misseldine, Southern Utah University 2015