Introduction to Applied Algebraic Systems

1: Modular Arithmetic 1.1: Sets, Functions, Numbers 1.2: Induction 1.3: Divisibility 1.4: Prime Numbers 1.5: Relations and Partitions 1.6: Modular Arithmetic 1.7: Equations in Zn 1.8: Bar Codes 1.9: The Chinese Remainder Theorem 1.10: Euler's -function 1.11: Theorems of Euler and Fermat 1.12: Public Key Cryptosystems 2: Rings and Fields 2.1: Basic Properties 2.2: Subrings and Subfields 2.3: Review of Vector Spaces 2.4: Polynomials 2.5: Polynomial Evaluation and Interpolation 2.6: Irreducible Polynomials 2.7: Construction of Finite Fields 2.8: Extension Fields 2.9: Multiplication Structure of Finite Fields 2.10: Primitive Elements 2.11: Subfield Structure of Finite Fields 2.12: Minimal Polynomials 2.13: Isomorphisms Between Fields 2.14: Error Correcting Codes 3: Groups and Permutations 3.1: Basic Properties 3.2: Subgroups 3.3: Permutation Groups 3.4: Matrix Groups 3.5: Even and Odd Permutations 3.6: Cayley's Theorem 3.7: Lagrange's Theorem 3.8: Orbits 3.9: Orbit/Stabilizer Theorem 3.10: Burnside's Theorem 3.11: K-Colourings 3.12: Cycle Index and Enumeration 4: Groups, Homomorphisms and Subgroups 4.1: Homomorphisms 4.2: The Isomorphism Theorems 4.3: Direct Products 4.4: Finite Abelian Groups 4.5: Conjugacy and the Class Equation 4.6: The Sylow Theorems 1 and 2 4.7: Sylow's Third Theorem 4.8: Solvable Groups 4.9: Nilpotent Groups 5: Rings and Polynomials 5.1: Homomorphisms and Ideals 5.2: Polynomial Rings 5.3: Divison Algorithm in F[x1, x2, ..., xn]; Single Divisor 5.4: Multiple Divisors; Groebner Bases 5.5: Ideals and Affine Varieties 5.6: Complex Numbers 5.7: Decomposition of Affine Varieties 5.8: Cubic Equations in One Variable 5.9: Parameters 5.10: Intersection Multiplicities 5.11: Singular and Nonsingular Points 6: Elliptic Curves 6.1: Elliptic Curves 6.2: Homogeneous Polynomials 6.3: Projective Space 6.4: Intersection of Lines and Curves 6.5: Defining Curves by Points 6.6: Classification of Conics 6.7: Reducible Conics and Cubics 6.8: The Nine Point Theorem 6.9: Groups on Elliptic Curves 6.10: The Arithmetic on an Elliptic Curve 6.11: Results Concerning the Structure of Groups on Elliptic Curves 7: Further Topics Relating to Elliptic Curves 418 7.1: Elliptic Curve Cryptosystems 7.2: Fermat's Last Theorem 7.3: Elliptic Curve Factoring Algorithm 7.4: Singular Curves of Form y2=x3+ax+b 7.5: Birational Equivalence 7.6: The Genus of a Curve 7.7: Pell's Equation Bibliography