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 Fields 2.8 Extension Fields 2.9 Multiplicative 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 The Cauchy-Frobenius Theorem 3.11 K-Colorings 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 4.10 The Enigma Encryption Machine 5 Rings and Polynomials 5.1 Homomorphisms and Ideals 5.2 Polynomial Rings 5.3 Division Algorithm in F[x1, x2, . . . , xn] : Single Divisor 5.4 Multiple Divisors: Groebner Bases 5.5 Ideals and Affine Varieties 5.6 Decomposition of Affine Varieties 5.7 Cubic Equations in One Variable 5.8 Parameters 5.9 Intersection Multiplicities 5.10 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 Related to Elliptic Curves 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 References Index