《Introduction To The Theory of Equations》

CHAPTER ⅠCOMPLEX NUMBERS1

1．Definitions．Operations with complex numbers1

2．Graphical representation of complex numbers．Addition5

3．Polar form of a complex number．Multiplication6

4．Demoivre＇s Theorem9

5．Roots of unity11

6．Primitive nth roots of unity14

7．Roots of complex numbers16

CHAPTER ⅡDIVISION AND FACTRIZATION OF POLYNOMIALS IN A FIELD20

8．Number-fields20

9．Fields of rational functions23

10．Polynomials in a field24

11．The division algorithm25

12．The Euclidean algorithm27

13．Greatest common divisor and least common multiple28

14．The identity AG+BF=D30

15．Subfields．Reducibility34

16．Unique Factorization Theorem36

CHAPTER ⅢFURTHER PROPERTIES OF POLYNOMIALS IN A FIELD40

17．Polynomials and equations having assigned roots40

18．Relations between roots and coefficients42

19．Derivative of a polynomial in an arbitrary field46

20．Repeated factors of a polynomial47

21．Synthetic division51

22．Taylor＇s Series52

23．Construction of polynomials having assigned properties55

CHAPTER ⅣTHEORY OF EQUATIONS IN THE FIELD OF RATIONAL NUMBERS59

24．A program for the study of the Theory of Equations59

25．Properties of integers59

26．Determination of rational roots60

27．Reducibility of polynomials64

CHAPTER ⅤTHEORY OF EQUATIONS IN THE FIELD OF REAL NUMBERS69

28．Introduction69

29．Ordered fields69

30．Compactness70

31．Continuity72

32．The fundamental property of continuous functions74

33．Rolle＇s Theorem75

34．Graphs of polynomials77

35．Bounds for real roots78

36．Isolation of the real roots of an equation with real coefficients80

37．Sturm＇s Theorem82

38．Budan＇s Theorem86

39．Descartes＇ Rule of Signs88

40．Horner＇s method91

41．Newton＇s method93

CHAPTER ⅥELIMINATION．RESULTANTS．SYMMETRIC FUNCTIONS97

42．Introduction97

43．Again the identity A(x)G(x)+B(x)F(x)=198

44．The resultant of two polynomials100

45．Factored form of the resultant101

46．Discriminant of a polynomial103

47．Symmetric functions105

48．Functional independence of the elementary symmetric functions105

49．The fundamental theorem on symmetric functions107

50．Degree and weight of a symmetric function108

51．Evaluation of symmetric functions110

52．The symmetric functiuns sk．Newton＇s identities114

53．Miacellaneous problems116

CHAPTER ⅦALGEBRAIC EXTENSIONS OF A FIELD119

54．Methods of extending a field119

55．Algebraic elements relative to a fidld119

56．Conjugate elements and conjugate fields120

57．Canonical form of the elements of R(α)．Primitive and imprimitive elements123

58．Multiple algebraic extensions of a field128

59．Radicals relative to a field134

60．Solution of the general cubic equation by radicals134

61．Trigonometric solution of the irreducible case137

62．Solution of the general quartic equation by radicals140

CHAPTER ⅧALGEBRAICALLY CLOSED FIELDS145

63．Introduction145

64．Proof of the Fundamental Theorem of Algebra145

65．Other algebraically closed fields150

CHAPTER ⅨCONSTRUCTIONS BY RULER AND COMPASSES154

66．Introduction154

67．The field R？ relative to R155

68．Constructible elements160

69．Irreducibility of the polynomial whose roots are the primitive nth roots of unity163

70．Inscribable regular polygons165

71．Construction of a regular polygon of 17 sides167

MISCELLANEOUS EXERCISES173

INDEX185