Abstract Algebra

A Gentle Introduction

Gary L. Mullen, James A. Sellers

• Publisher: CRC Press
• Year: 2016
• Print ISBN: 9781032476971
• eBook ISBN: 9781482250091
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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

Chapter 1

Elementary Number Theory

1.1 Divisibility

1.2 Primes and factorization

1.3 Congruences

1.4 Solving congruences

1.5 Theorems of Fermat and Euler

1.6 RSA cryptosystem

This chapter deals with several fundamental topics from elementary number theory. The reader will find that many of these topics will later be generalized and discussed in other chapters such as the group, ring, and field chapters.

1.1 Divisibility

We begin by discussing the notion of divisibility of integers. Indeed, this is one of the oldest concepts in number theory. The topic has been studied in depth for at least 2000 years. In fact, Euclid (∼325BC–265BC) dealt with the topic of divisibility in Book VII of his Elements.

Let a be an integer and let d be an integer. We say that d divides a if there is an integer k so that a = dk. The important point here is that k must be an integer. If there are such integers, we denote the fact that d divides a by using the notation d|a.

For example, 2|8 since 8 = 2(4); 36|108 since 108 = 36(3); 3|(−36) since −36 = 3(−12); and for any integer m, 3|(15m + 3) since 15m + 3 = 3(5m + 1). On the other hand, 3 does not divide 13 as there is no integer k with 13 = 3k.

The reader should be aware that d|a is just a notation for the fact that the integer d divides a. Be careful, do not confuse this with the notation d/a which, as usual, represents the value d divided by a.

In the following lemma, we provide a few basic properties involving the divisibility of integers.

Lemma 1.1 Let a, b, d be integers with d > 0.

• (i) If d|a and d|b, then d|(a + b);
• (ii) If d|a and d|b, then d|(a − b);
• (iii) If d|a and d|b, then d|(ma + nb) for any integers m and n;
• (iv) If m|d and d|a, then m|a.

Proof: To prove part (i), we may assume that a = dk and b = dl where k and l are integers. Hence a + b = dk + dl = d(k + l) so that d divides a + b (since k + l is an integer).

The proof of part (ii) is similar and hence omitted. Proofs of the remaining parts are left to the exercises.

We now state two extremely important results about positive integers. These results are known as the Division and Euclidean Algorithms, respectively. The reader should not be confused or worried about the term “algorithm;” it is simply a term that is used to indicate a method to compute something. While mathematicians often use this terminology, it is also very heavily used by people working in the field of computer science.

Theorem 1.2 (Division Algorithm) Let a and b be two positive integers. Then there are integers q and r so that b = aq + r where 0 ≤ r < a.

A proof of this result is given in the Appendix, Section 2 where we discuss the Well-Ordering Principle. For the moment, let’s just use this result.

The integers q and r are often called the quotient and the remainder, respectively.

This terminology comes from long division. For example, consider the integers a = 4 and b = 23. Dividing 23 by 4, we see that 23 = 5(4) + 3. Hence from the Division Algorithm the quotient q is 5 and the remainder r is 3.

Note ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Elementary Number Theory
• 2 Groups
• 3 Rings
• 4 Fields
• 5 Finite Fields
• 6 Vector Spaces
• 7 Polynomials
• 8 Linear Codes
• 9 Appendix
• 10 Hints and Partial Solutions to Selected Exercises
• Bibliography
• Index