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2nd Edition

Introduction to Abstract Algebra




ISBN 9781498731614
Published October 23, 2015 by Chapman and Hall/CRC
352 Pages 52 B/W Illustrations

 
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Book Description

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.

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
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

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Author(s)

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.

Reviews

"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

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