The study of “abstract algebra” grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted.
For example, we are familiar with the notion that real numbers are closed under multiplication and division (that is, if we add or multiply a real number, we get a real number). But if we divide one integer by another integer, we may not get an integer as a result—meaning that integers are not closed under division. We also know that if we take any two integers and multiply them in either order, we get the same result—a principle known as the commutative principle of multiplication for integers. By contrast, matrix multiplication is not generally commutative. Students of abstract algebra are interested in these sorts of properties, as they want to determine which properties hold true for any set of mathematical objects under certain operations and which types of structures result when we perform certain operations. Abstract algebra has applications in a variety of diverse fields, including computation, physics, and economics and, as a result, is an important area in mathematics.
We will begin this course by reviewing basic set theory, integers, and functions in order to understand how algebraic operations arise and are used. We then will proceed to the heart of the course, which is an exploration of the fundamentals of groups, rings, and fields.
Upon successful completion of this course, the student will be able to:
- Describe and generate groups, rings, and fields.
- Relate abstract algebraic constructs to more familiar number sets and operations and see from where the constructs derive.
- Identify examples of specific constructs.
- Identify and differentiate between different structures and understand how changing properties give rise to new structures.
- Explain the theory behind relations and functions and identify domains and images of functions, based on the structures given.
- Explain how functions may relate seemingly dissimilar structures to each other and how knowing properties of one structure allows us to know the same properties in the related structure, if certain functions exist between them.
