The field of algebra, discrete mathematics, and number theory encompasses one of the primary branches of pure mathematics. Problems in this field often arise (or follow naturally from) a problem that is easily stated involving counting, divisibility, or some other basic arithmetic operation. While many of the problems are easily stated, the techniques used to attack these problems are some of the most difficult and advanced in mathematics. Algebra, discrete mathematics, and number theory have seen somewhat of a renaissance in the past couple of decades with Andrew Wiles' proof of Fermat's Last Theorem, the increasing need for more advanced techniques in cryptography and coding theory arising from the internet, as well as surprising applications in areas such as particle physics and mathematical biology. Algebra, discrete mathematics, and number theory have been featured in the motion picture Good Will Hunting, the play Fermat's Last Tango, as well as numerous episodes of the CBS hit drama Numb3rs.
The core courses of an algebra, discrete mathematics, and number theory concentration are matrix analysis (8530) and abstract algebra I and II (8510-8520). Matrix analysis is a basic course in linear algebra dealing with topics such as similarity of matrices, eigenvalues, and canonical forms just to name a few. Abstract algebra I and II abstract the familiar structures of the integers, rational numbers, matrices, etc. into the concepts of groups, rings, fields, and modules. One also studies one of the crowning achievements of the subject, Galois theory. In addition to the department's broad course requirements, it is expected a student in algebra, discrete mathematics, and number theory will gain a deeper level of understanding of each of the concentrations listed below as well as taking significant advanced courses in that student's particular concentration.