This course presents an axiomatic approach to the study of algebraic systems. It begins by investigating the most fundamental concepts behind integer arithmetic. It then shows how all other arithmetic operations involving integers are justified from these basic concepts which are called postulates. Other topics involving integers such as proof by induction, divisibility, congruence, and modular arithmetic are also discussed. A general discussion of algebraic systems such as groups, rings, integral domains, and fields includes the tools used to analyze algebraic systems such as sets, mappings between sets, relations defined on sets, permutations, homomorphisms and isomorphisms. These tools are used to compare algebraic systems defined on sets of integers, rational, real, and complex numbers. Examples involving matrices, coding theory and applications to computer science are used to illustrate the concepts.
