CHAPTER Ⅰ COMPLEX NUMBERS 1
1．Definitions．Operations with complex numbers 1
2．Graphical representation of complex numbers．Addition 5
3．Polar form of a complex number．Multiplication 6
4．Demoivre＇s Theorem 9
5．Roots of unity 11
6．Primitive nth roots of unity 14
7．Roots of complex numbers 16
CHAPTER Ⅱ DIVISION AND FACTRIZATION OF POLYNOMIALS IN A FIELD 20
8．Number-fields 20
9．Fields of rational functions 23
10．Polynomials in a field 24
11．The division algorithm 25
12．The Euclidean algorithm 27
13．Greatest common divisor and least common multiple 28
14．The identity AG+BF=D 30
15．Subfields．Reducibility 34
16．Unique Factorization Theorem 36
CHAPTER Ⅲ FURTHER PROPERTIES OF POLYNOMIALS IN A FIELD 40
17．Polynomials and equations having assigned roots 40
18．Relations between roots and coefficients 42
19．Derivative of a polynomial in an arbitrary field 46
20．Repeated factors of a polynomial 47
21．Synthetic division 51
22．Taylor＇s Series 52
23．Construction of polynomials having assigned properties 55
CHAPTER Ⅳ THEORY OF EQUATIONS IN THE FIELD OF RATIONAL NUMBERS 59
24．A program for the study of the Theory of Equations 59
25．Properties of integers 59
26．Determination of rational roots 60
27．Reducibility of polynomials 64
CHAPTER Ⅴ THEORY OF EQUATIONS IN THE FIELD OF REAL NUMBERS 69
28．Introduction 69
29．Ordered fields 69
30．Compactness 70
31．Continuity 72
32．The fundamental property of continuous functions 74
33．Rolle＇s Theorem 75
34．Graphs of polynomials 77
35．Bounds for real roots 78
36．Isolation of the real roots of an equation with real coefficients 80
37．Sturm＇s Theorem 82
38．Budan＇s Theorem 86
39．Descartes＇ Rule of Signs 88
40．Horner＇s method 91
41．Newton＇s method 93
CHAPTER Ⅵ ELIMINATION．RESULTANTS．SYMMETRIC FUNCTIONS 97
42．Introduction 97
43．Again the identity A(x)G(x)+B(x)F(x)=1 98
44．The resultant of two polynomials 100
45．Factored form of the resultant 101
46．Discriminant of a polynomial 103
47．Symmetric functions 105
48．Functional independence of the elementary symmetric functions 105
49．The fundamental theorem on symmetric functions 107
50．Degree and weight of a symmetric function 108
51．Evaluation of symmetric functions 110
52．The symmetric functiuns sk．Newton＇s identities 114
53．Miacellaneous problems 116
CHAPTER Ⅶ ALGEBRAIC EXTENSIONS OF A FIELD 119
54．Methods of extending a field 119
55．Algebraic elements relative to a fidld 119
56．Conjugate elements and conjugate fields 120
57．Canonical form of the elements of R(α)．Primitive and imprimitive elements 123
58．Multiple algebraic extensions of a field 128
59．Radicals relative to a field 134
60．Solution of the general cubic equation by radicals 134
61．Trigonometric solution of the irreducible case 137
62．Solution of the general quartic equation by radicals 140
CHAPTER Ⅷ ALGEBRAICALLY CLOSED FIELDS 145
63．Introduction 145
64．Proof of the Fundamental Theorem of Algebra 145
65．Other algebraically closed fields 150
CHAPTER Ⅸ CONSTRUCTIONS BY RULER AND COMPASSES 154
66．Introduction 154
67．The field R？ relative to R 155
68．Constructible elements 160
69．Irreducibility of the polynomial whose roots are the primitive nth roots of unity 163
70．Inscribable regular polygons 165
71．Construction of a regular polygon of 17 sides 167
MISCELLANEOUS EXERCISES 173
INDEX 185