Math 135

#class

Algebra for Honours Mathematics

That's for you guys to figure out; I'm too old
- Prof. Wentang Kuo

  1. Introduction to the Language of Mathematics
  2. Logical Analysis of Mathematical Statements
  3. Proving Mathematical Statements
  4. Mathematical Induction
    1. Notation for Summations, Products, and Recurrences
    2. Proof by Induction
    3. The Binomial Theorem
    4. Proof by Strong Induction
  5. Set Theory
  6. The Greatest Common Divisor
    1. The Division Algorithm
    2. The Greatest Common Divisor
    3. Certificate of Correctness and Bézout's lemma
    4. Extended Euclidean Algorithm
    5. Further Properties of the Greatest Common Divisor
    6. Prime Numbers
    7. Prime Factorizations and the Greatest Common Divisor
  7. Linear Diophantine Equations
    1. The Existence of Solutions in Two Variables
    2. Finding all Solutions in Two Variables
  8. Congruency and Modular Arithmetic
    1. Congruence
    2. Elementary Properties of Congruence
    3. Congruence and Remainders
    4. Linear Congruences
    5. Non-Linear Congruences
    6. Congruence Classes and Modular Arithmetic
    7. Fermat's Little Theorem
    8. Chinese Remainder Theorem
    9. Splitting Modulus Theorem
  9. The RSA Public-Key Encryption Scheme
    1. Public Key Cryptography
    2. Implementing the RSA Scheme
    3. Proving that the RSA Scheme Works
  10. Complex Numbers
    1. Standard Form
    2. Conjugate and Modulus
    3. The Complex Plane and Polar Form
    4. De Moivre's Theorem
    5. Complex nth Roots
    6. Square Roots and the Quadratic Formula
  11. Polynomials
    1. Introduction
    2. Arithmetic with Polynomials
    3. Roots of Complex Polynomials and the Fundamental Theorem of Algebra
    4. Real Polynomials and the Conjugate Roots Theorem
    5. Integer Polynomials and the Rational Roots Theorem
  12. Additional Material
    1. Prime Numbers and the Riemann Hypothesis