FUNDAMENTAL CONCEPTS OF ALGEBRA

FUNDAMENTAL CONCEPTS OF ALGEBRA

Donald L. White Department of Mathematical Sciences

Kent State University Release 3.0

January 12, 2009

Copyright c 2009 by D. L. White

Contents

1 Number Systems

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1.1 The Basic Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Algebraic Properties of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Formal Constructions of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Basic Number Theory

35

2.1 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Divisibility of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Division Algorithm and Greatest Common Divisor . . . . . . . . . . . . . . . . . . . 44

2.4 Properties of the Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Prime Factorizations and Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.7 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.8 Congruence and Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Polynomials

84

3.1 Algebraic Properties of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2 Binomial Coefficients and Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Divisibility and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.4 Synthetic Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5 Factors and Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.7 Irreducible Polynomials as Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A Trigonometry Review

139

B Answers to Selected Problems

141

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Chapter 1

Number Systems

In this chapter we study the basic arithmetic and algebraic properties of the familiar number systems the integers, rational numbers, real numbers, and the possibly less familiar complex numbers. We will consider which algebraic properties these number systems have in common as well as the ways in which they differ.

We will use the following notation to denote sets of numbers.

N = {1, 2, 3, . . .} = Natural Numbers

Z = {0, ?1, ?2, ?3, . . .} = Integers

Q=

a b

a, b Z, b = 0

= Rational Numbers

R = Real Numbers

C = Complex Numbers

1.1 The Basic Number Systems

The first numbers anyone learns about are the "counting numbers" or natural numbers 1, 2, 3, . . ., which we will denote by N. We eventually learn about the basic operations of addition and multiplication of natural numbers. These operations are examples of binary operations, that is, operations that combine any two natural numbers to obtain another natural number. Addition and multiplication of natural numbers satisfy some very nice properties, such as commutativity, associativity, and the distributive law, which we will study more formally in a later section.

The other familiar arithmetic operations of subtraction and division are really just the "inverse operations" of addition and multiplication, and will not be considered as basic operations. (Although multiplication of natural numbers is really just repeated addition, this is a much less obvious interpretation in other number systems.) If we only wish to consider the natural numbers, we quickly encounter problems with subtraction and division. These operations can be performed on pairs of natural numbers only in some cases. For example, 3 - 5 and 3 ? 5 are not natural numbers.

In order to be able to subtract, we introduce the number 0 and the "negatives" of the natural numbers to obtain the set of integers Z = {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}. The number 0 acts as a

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