Abstract Algebra: A Gentle Introduction

ISBN-13: 9781032476971 / Angielski / Miękka / 2023 / 214 str.

Gary L. Mullen; James A. Sellers

Opis

This book introduces the basic notions of abstract algebra to sophomores and perhaps even junior mathematics majors who have a relatively weak background with conceptual courses. It introduces the material with many concrete examples and establishes a firm foundation for introducing more abstract mathematical notions.

 

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

Spis treści

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

O autorze

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.