## Course Syllabus for "MA231: Abstract Algebra I"

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.

### Learning Outcomes

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.

### Course Requirements

In order to take this course, you must:

√ Have access to a computer.

√ Have continuous broadband Internet access.

√ Have the ability/permission to install plug-ins or software (e.g. Adobe Reader, QuickTime, and Flash viewer).

√ Be competent in the English language.