Fundamental Concepts of Algebra

Uncommonly interesting introduction illuminates complexities of higher mathematics while offering a thorough understanding of elementary mathematics. Covers development of complex number system and elementary theories of numbers, polynomials and operations, determinants, matrices, constructions and graphical representations. Several exercises without solutions.

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