University College Roosevelt Academy Middelburg - Science

SCI 312 Advanced Mathematics

Content

This course is meant as an introduction to the fascinating field of abstract algebra; it will introduce some of the fundamental algebraic systems that are both interesting and of wide use. The aim of this course is to arrive at some significant results in each of these systems. The starting point is the following: we have a collection of objects and we assume that we can combine the elements of this set to obtain again the elements of this set; combining the elements of the set is called an operation. Then we condition or regulate the nature of the set by imposing rules on how these operations behave on the set; these axioms define the particular structure on the set. In the course of time, it was noticed that there were many concrete mathematical systems that satisfied these axioms. In this course, we will study some of the basic axiomatic algebraic systems: groups, rings and fields. A group can be described by a closed set of elements equipped with an operation (multiplication) that is associative, it has an identity element and an inverse element for each element of the set. A ring is a closed set of elements under two operations (multiplication and addition), it is associative for both operations, commutative for addition, and distributive. Furthermore, there is a zero element and an additive inverse element for each element of the set. Finally, a field is a commutative ring in which every nonzero element has a multiplicative inverse; a field allows for the operations addition, subtraction, multiplication and division.