Introduction To The Theory of EquationsPDF电子书下载

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