Fundamental Concepts of Algebra

CHAPTER 1. OUR NUMBER SYSTEM 1-1 Sets 1-2 Cardinal numbers 1-3 Equivalence relations 1-4 Peano's postulates 1-5 Addition and multiplication 1-6 Order relations 1-7 Inverses 1-8 Positive raional numbers 1-9 Negative numbers 1-10 Real numbers *1-11 Postulates for real numbers 1-12 Properties of real numbers 1-13 Transfinite cardinal numbers 1-14 Group; number system 1-15 Complex numbers 1-16 Properties of complex numbers 1-17 De Moivre's Theorem *1-18 Fields and number systems CHAPTER 2. THEORY OF NUMBERS 2-1 Divisibility 2-2 Division Algorithm 2-3 Prime numbers 2-4 Unique Factorization Theorem 2-5 Euclidean Algorithm 2-6 Bases 2-7 Decimal notation *2-8 Congruences *2-9 Residue classes. Euler's F-function *2-10 Evaluation of F(m) *2-11 Linear congruences *2-12 Diophantine problems CHAPTER 3. THEORY OF POLYNOMIALS 3-1 Polynomials 3-2 Rings of polynomials 3-3 Rational functions 3-4 Divisibility 3-5 Division Algorithm 3-6 Irreducible polynomials 3-7 Euclidean Algorithm 3-8 Change of variable *3-9 Ideals 3-10 Functions 3-11 Limits 3-12 Continuity 3-13 Continuous functions 3-14 Derivatives *3-15 Taylor's series *3-16 Analytic functions CHAPTER 4. THEORY OF EQUATIONS 4-1 Zeros of a polynomial 4-2 Synthetic division 4-3 Change of variable 4-4 Number of roots 4-5 Determination of the roots 4-6 Conjugate imaginary roots 4-7 Elementary symmetric polynomials 4-8 Transformations of roots *4-9 Cubic equations *4-10 Quartic equations 4-11 Descartes' Rule of Signs 4-12 Sturm's Theorem 4-13 Multiple roots 4-14 Approximate solutions CHAPTER 5. DETERMINANTS AND MATRICES 5-1 Historical development 5-2 Matrices 5-3 Permutations 5-4 Inversions 5-5 Transpositions 5-6 Even and odd permutations 5-7 Determinants 5-8 Properties of determinants 5-9 Expansion of determinants 5-10 Minors 5-11 Cramer's Rule 5-12 Systems of linear equations 5-13 Linear dependence 5-14 Applications in analytic geometry 5-15 Geometric transformations CHAPTER 6.CONSTRUCTIONS 6-1 Classical constructions 6-2 Elementary classical constructions 6-3 The algebraic viewpoint 6-4 Basic classical constructions 6-5 Construction of roots of equations 6-6 Famous construction problems 6-7 Nonclassical geometric trisections 6-8 Mechanical angle trisectors 6-9 Linkages 6-10 Summary CHAPTER 7. GRAPHICAL REPRESENTATIONS 7-1 Euclidean and complex spaces 7-2 Polynomials 7-3 Conic sections 7-4 Quadric surfaces 7-5 Higher plane surves 7-6 Rational functions 7-7 Algebraic functions 7-8 Curve tracing 7-9 Special graphs 7-10 Graphical solutions 7-11 Curve fitting 7-12 Conclusion BIBLIOGRAPHY SYMBOLS AND NOTATIONS INDEX