Algebra and its Applications

COURSE DESCRIPTION

This course is a first introduction to structures of abstract algebra, specifically, groups, rings, and fields. The first quarter of the course will be devoted to elementary number theory. Then, we will focus on group theory. This is the core of this class, and we will devote half of the course to this topic. We will end the class with an overview on rings and fields.

This class is proof-based. However, it is not expected that you have taken a proof-based course before, and there will be an opportunity to learn and improve proof-writing.

In addition, there will be a significant emphasis on real-life problems which are modeled by abstract algebraic structures.

There will be a final project on applications of group theory (e.g. cryptography, codes, music, physics, and Rubik’s cube).

Prerequisites

  • Basic knowledge of solution methods for linear equations
  • Familiarity with basic properties of sets and functions
COURSE GOALS

On completion of the course, students will:

  • be familiar with concepts and techniques in group theory;
  • be able to identify real-life problems that can be modeled with abstract algebra;
  • be familiar with mathematical formality and rigor.
COURSE CONTENTS
  1. Basic number theory (4 weeks)
    1. Postulates for the integers
    2. Mathematical induction
    3. Divisibility
    4. Prime factors and greatest common divisors
    5. Congruence of integers
    6. Arithmetic with congruence classes
  2. Group theory (5 weeks)
    1. Definition of a group
    2. Properties of group elements
    3. Subgroups
    4. Cyclic groups
    5. Homomorphisms
    6. Isomorphisms
    7. Finite permutation groups
    8. Cayley's Theorem
    9. Cosets of a subgroup
    10. Normal subgroups
    11. Quotient groups
    12. Direct sums
  3. Rings and Fields (4 weeks)
    1. Definition of a ring
    2. Integral domains and fields
    3. The field of quotients of an integral domain
    4. Ideals and quotient rings
    5. Ring homomorphisms

Bibliography

  1. Gilbert, Linda; Gilbert, Jimmie (2015). Elements of Modern Algebra, Eighth Edition, Cengage.
  2. Beachy, John A.; Blair, William D. (2005). Abstract Algebra, Third Edition, Waveland Press.
  3. Adhikari, Mahima R.; Adhikari, Avishek (2014). Modern Algebra with Applications, Springer.

Support Sessions

2 hours per week with a teaching assistant

Grading

Midterm exam (25%), final exam (40%), homework (15%), project (20%)