THeory of Equations - TABLE OF CONTENTS

Chapter 1: Real and Complex Polynomials (College algebra): 1.1 The Real Number System; 1.2 A little History of Real and Complex Numbers; 1.3 Complex Numbers; 1.4 Polynomial arithmetic; 1.5 Degree of polynomials; 1.6 The Division Algorithm; 1.7 Factors and roots; 1.8 Greatest common divisors and Euclidean algorithm; 1.9 Unique Factorization; 1.10 Formal Differentiation of Polynomials; 1.11 Test for multiple roots; 1.12 Partial Fraction Decomposition; 1.13 The Resultant
Chapter 2: Numerical solution of polynomial equations (Numerical Analysis): 2.1 Numerical Algorithms; 2.2 Evaluation of polynomials (synthetic division); 2.3 Taylor's Series and Horner's Process; 2.4 Newton's upper bound on modulus of roots; 2.5 Graphing; 2.6 Descartes' Rule of signs; 2.7 Bisection Method; 2.8 Horner's Method; 2.9 Fixed Point Iteration method; 2.10 Newton's Method, real case; 2.11 Newton's Method, complex case; 2.12 Newton Barstow Algorithm; 2.13 Other root finding algorithms; 2.14 Polynomial Interpolation
Chapter 3: The Fundamental Theorem of Algebra (Topology): 3.1 History; 3.2 Gauss's Fourth Proof; 3.3 Topological Proof; 3.4 Analytical proof; 3.5 Another Curve Proof; 3.6 Connection between Gauss's Proof and Newton's method; 3.7 Where Newton's method does not converge; 3.8 Real Newton's method revisited; 3.9 Iteration of quadratic polynomials.
Chapter 4: Exact Solutions (Ancient and Modern Algebra): 4.1 Solutions of Quadratic Equations; 4.2 Omar Khayyam and Viete's solution of the cubic.; 4.3 History of the Cubic and Biquadratic; 4.4 Algebraic Solution of the Cubic; 4.5 Solution of the Biquadratic Equation; 4.6 Newton's Identities; 4.7 More on Newton's Identities (optional); 4.8 Symmetric Polynomials; 4.9 Lagrange's Solution of the Biquadratic; 4.10 Insolvability of the Quintic
Chapter 5: Factoring integer polynomials (Number Theory): 5.1 Rational Polynomials and Algebraic Numbers; 5.2 Polynomials with integer coefficients; 5.3 Rational Roots and Factors; 5.4 Eisenstein's Irreducibility Criterion; 5.5 Elementary Factoring Methods; 5.6 Modular Arithmetic; 5.7 Polynomials over Z/p; 5.8 Factoring in Z/p[x]; 5.9 Comparison of factoring Integers and Polynomials; 5.10 Roots of Polynomials in Z/n
Chapter 6: Elliptic Functions (Analysis): 6.1 Trigonometric and Hyperbolic Functions; 6.2 The Historical Background; 6.3 The Jacobi Elliptic Functions; 6.4 The inverse Jacobi Elliptic Functions; 6.5 Elliptic Integrals; 6.6 The Complex Theory
Chapter 7: Polynomials of several variables (Computational Algebra): 7.1 Multivariable Polynomials; 7.2 Term orders; 7.2 Ideals and Varieties; 7.3 The reduction process; 7.4 Grobner Bases, the normal form; 7.5 S-polynomials and Buchberger's Algorithm; 7.6 Algebraic Solution of Systems of polynomial Equations

Table of Contents