# Mathematics 423

MATH 423: Concepts of Abstract Algebra

Prerequisites: MATH 300

Credit Hours: (3)

A study of the structure of algebraic systems.

Detailed Description of Content of Course

The major topics covered in this course are those which represent the foundation of the field of modern algebra. It is, for most students, an introduction to the axiomatic method. Topics included in MATH 423 are:

a. Equivalence relations
b. Binary operations
c. Groups
d. Subgroups
e. Cyclic Groups
f. Cosets
g. Homomorphisms
h. Factor groups
i. Rings
j. Integral domains
k. Polynomial rings
l. Fields
j. Finite Fields

The history of the main results covered in the course will also be discussed.

Detailed Description of Conduct of Course

Most instructors will present the course material in a lecture format. Students may be required to prepare and present problems for class discussion.

Goals and Objectives of the Course

Abstract algebra has become one of the three main divisions of mathematics. This course offers an introduction to that area as well as an introduction to the axiomatic method so important to modern mathematics. Students who finish the course should:

1. be better able to understand and write mathematical proofs;
2. be able to determine whether a set with a binary operation is a group;
3. be able to determine whether a subset of a group is a subgroup;
4. know the concept of a cyclic generator and be able to determine whether a group is cyclic;
5. know and be able to apply basic theorems on groups, including Cauchy’s Theorem;
6. be able to determine whether a set with two binary operations is a ring;
7. know and be able to apply basic theorems on rings;
8. be able to determine whether a given ring is a field;
9. know and be able to apply basic theorems on fields, including finite fields.

The course will examine applications of abstract algebra as time permits.

Assessment Measures

Graded instruments may include in-class tests, homework assignments, pop quizzes, presentation of problem assignments and class participation.

Other Course Information

This course may require library research in recent developments in the field. This course is intended for students who will teach or pursue graduate studies.

Review and Approval

Revised 2013

Sept. 2001 Review Stephen Corwin, Chair