Overview1. FUNDAMENTALS. Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations. 2. THE INTEGERS. Postulates for the Integers (Optional). Mathematical Induction. Divisibility. Prime Factors and Greatest Common Divisor. Congruence of Integers. Congruence Classes. Introduction to Coding Theory (Optional). Introduction to Cryptography (Optional). 3. GROUPS. Definition of a Group. Properties of Group Elements. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms. 4. MORE ON GROUPS. Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Cosets of a Subgroup. Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional). 5. RINGS, INTEGRAL DOMAINS, AND FIELDS. Definition of a Ring. Integral Domains and Fields. The Field of Quotients of an Integral Domain. Ordered Integral Domains. 6. MORE ON RINGS. Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring. Maximal Ideals (Optional). 7. REAL AND COMPLEX NUMBERS. The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers. 8. POLYNOMIALS. Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in _F[x]_ . Zeros of a Polynomial. Solution of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field.