Abstract Algebra: A Gentle Introduction

by Gary L. Mullen, James A. Sellers

Overview

Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra.

Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors

who need to have an introduction to the topic.

As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a "gentle introduction," meant to provide enough for a course, yet also enough to encourage them toward future study of the topic.

Features

• Groups before rings approach



• Interesting modern applications



• Appendix includes mathematical induction, the well-ordering principle, sets, functions, permutations, matrices, and complex nubers.



• Numerous exercises at the end of each section



• Chapter "Hint and Partial Solutions" offers built in solutions manual

Product Details

ISBN-13:	9781032476971
Publisher:	CRC Press
Publication date:	01/21/2023
Series:	Textbooks in Mathematics
Pages:	214
Product dimensions:	5.90(w) x 8.90(h) x 0.60(d)

About the Author

Gary Mullen is Professor of Mathematics, The Pennsylvania State University, where he earned his Ph.D. His main interest is finite fields, and is founder of the journal "Finite Fields and Their Introduction." He is also the Editor of The Handbook of Finite Fields published by CRC Press.

James Sellers is Professor and Associate Head for Undergraduate Mathematics, The Pennsylvania State University, where he also earned his Ph.D. He has published many research articles and won awards related to his efforts to advance mathematics education.

Table of Contents

Elementary Number Theory

Divisibility

Primes and factorization

Congruences

Solving congruences

Theorems of Fermat and Euler

RSA cryptosystem

Groups

De nition of a group

Examples of groups

Subgroups

Cosets and Lagrange's Theorem

Rings

Defiition of a ring

Subrings and ideals

Ring homomorphisms

Integral domains

Fields

Definition and basic properties of a field

Finite Fields

Number of elements in a finite field

How to construct finite fields

Properties of finite fields

Polynomials over finite fields

Permutation polynomials

Applications

Orthogonal latin squares

Die/Hellman key exchange

Vector Spaces

Definition and examples

Basic properties of vector spaces

Subspaces

Polynomials

Basics

Unique factorization

Polynomials over the real and complex numbers

Root formulas

Linear Codes

Basics

Hamming codes

Encoding

Decoding

Further study

Exercises

Appendix

Mathematical induction

Well-ordering Principle

Sets

Functions

Permutations

Matrices

Complex numbers

Hints and Partial Solutions to Selected Exercises

Editorial Reviews

As the subtitle implies, those seeking a standard undergraduate text in abstract algebra should look elsewhere. The authors provide readers with a very brief introduction to some of the central structures of algebra: groups, rings, fields, and vector spaces. As an example of the text's brevity, its treatment of groups consists of definitions, examples, and a discussion of subgroups and cosets that culminates in LaGrange's theorem. There is no mention of group homomorphisms, normal subgroups, or quotient groups. Nonetheless, various applications of the subject not often addressed in traditional texts are treated within this work. It appears that the intent is to provide enough content for readers to comprehend these applications. Just enough elementary number theory is presented to allow a discussion of the RSA cryptosystem. Sufficient material on finite fields is given for a discussion of Latin squares and the Diffie-Hellman public key exchange. Adequate linear algebra topics foster a discussion of Hamming codes. This text will be suitable for an algebra-based course introducing students to abstract mathematical thought or an algebra course with an emphasis on applications.

—D. S. Larson, Gonzaga University, Choice magazine 2016

As the subtitle implies, those seeking a standard undergraduate text in abstract algebra should look elsewhere. The authors provide readers with a very brief introduction to some of the central structures of algebra: groups, rings, fields, and vector spaces. As an example of the text's brevity, its treatment of groups consists of definitions, examples, and a discussion of subgroups and cosets that culminates in LaGrange's theorem. There is no mention of group homomorphisms, normal subgroups, or quotient groups. Nonetheless, various applications of the subject not often addressed in traditional texts are treated within this work. It appears that the intent is to provide enough content for readers to comprehend these applications. Just enough elementary number theory is presented to allow a discussion of the RSA cryptosystem. Sufficient material on finite fields is given for a discussion of Latin squares and the Diffie-Hellman public key exchange. Adequate linear algebra topics foster a discussion of Hamming codes. This text will be suitable for an algebra-based course introducing students to abstract mathematical thought or an algebra course with an emphasis on applications.

—D. S. Larson, Gonzaga University, Choice magazine, 2016

From the Publisher