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Contents ; 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: 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: Burnside’s Theorem; 3.11: K-Colourings; 3.12; 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: Division 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: 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

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