Course MAT340

Applied Algebra

Algebra, created in the medieval era in the Arab world, is the study of operations (such as addition and multiplication) on objects (such as numbers, polynomials, and matrices).  Abstract algebra, developed in the 19th century, studies common features and properties of such operations on algebraic systems.  Algebra is a fundamental and universal language upon which most other branches of mathematics are built.  In recent decades, algebra has become an increasingly important field of study due to its numerous contemporary applications in physics, chemistry, computer science, data communication and security.

We start with properties of integers and modular arithmetic, and then move on to the theory of groups, rings and fields.  Groups are motivated by the study of symmetry and are important in crystallography, quantum physics and cryptography.  Rings and fields are abstractions of standard notions of addition and multiplication, with applications to error-correcting codes.

Abstract algebra is a contemporary subject as its concepts and methodologies are used by working mathematicians, computer scientists, physicists and chemists.  Faculty from other departments will be invited to give guest lectures on applications of algebraic structures.  This course will therefore serve as a bridge course to further studies in many departments, and it will expose students to potential junior paper or senior thesis projects through dialogues with guest lecturers from other departments.

Topics

We will use the textbook Contemporary Abstract Algebra by Joseph Gallian covering the following topics:

  • properties of integers and modular arithmetic, Euclidean algorithm, factorization, the Chinese remainder theorem
  • Groups, subgroups and their properties, cyclic groups and public-key encryption schemes
  • Symmetry groups and their manifestation in nature, crystallographic groups, physical properties of molecules such as fullerene or other objects with symmetric structures
  • Isomophisms and automorphisms of groups, Cayley's theorem, Lagrange's theorem, Fermat's Little Theorem and its role in polynomial-time primality testing algorithms
  • Rings, ideals, fields and their properties
  • Polynomial rings, vector spaces, extension fields
  • Finite fields, Berlekamp's algorithm for factoring polynomials over finite fields, Hamming code, linear code and error-correcting codes such as Reed-Solomon codes
Description of classes

The class meets twice per week for 90 minutes.

Notes
  • The course will cover key aspects of the algebra sequence MAT345/346, but will place an emphasis on applications and algorithms.
  • It will run concurrently with MAT345 in the Fall.
  • Intended as a more applied alternative to MAT345 for students majoring in mathematics or other disciplines.  Students cannot take both MAT340 and MAT345 for University credit.
Who Takes This Course
  • Intented for students, mostly sophomores and juniors, with a serious interest in applied mathematics
  • Students who plan to go to graduate school to study algebra, number theory, algebraic geometry should take MAT345/346 instead.
Placement and Prerequisites

The course assumes a good working knowledge of linear algebra.  Students should have taken MAT202 or 204 or 217 or the equivalent.