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Chapter 1: Integers

Introduction

 
Chapter 1 of the text introduces the basic ideas from number theory that are a prerequisite to studying abstract algebra. Many of the concepts introduced in this chapter can be abstracted to much more general situations. For example, in Chapter 3 of the text you will be introduced to the concept of a group. One of the first broad classes of groups that you will meet depends on the definition of a cyclic group, one that is obtained by considering all powers of a particular element. The examples in Section 1.4, constructed using congruence classes of integers, actually tell you everything you will need to know about cyclic groups. In fact, although Chapter 1 is very concrete, it is a significant step forward into the realm of abstract algebra.

This may be an appropriate place to give a short (but rather glib) definition of the term "abstract algebra". Let's begin with a short definition of "algebra". The step from arithmetic to algebra occurs when we begin to use variables in place of numbers, so the short definition is that if you see an "x" it must be algebra. The operations we use for variables are still the usual operations for numbers: addition, subtraction, multiplication, and division. But soon we need to talk about these operations for polynomials, and then perhaps for matrices, when we study how to solve systems of linear equations.

In "abstract" algebra we allow the variables x and y to represent elements other than just numbers, such as polynomials and matrices. We also allow the operation to be a variable, so in Chapter 3 you will see expressions like x*y, where the * represents an operation we want to use for the variables x and y. The short definition of "abstract" algebra is that we allow variables to represent the operations, as well as the elements we want to use.

Before beginning to study this chapter, we recommend that you read our general advice, in the section To the Student on pages ix - xviii of the textbook. You will find some somments about logic, and how to get started writing your own proofs. If you would like more detailed help, we recommend these links on logic and proofs and on the history of algebra.


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