This course is a first introduction to structures of abstract algebra, specifically, groups, rings, and fields. There will be a significant emphasis on real-life problems which are modeled by abstract algebraic structures. 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. Applications will be discussed within each topic.

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). Project suggestions will be introduced during the first half of the semester.

Prerequisites

In addition to the common requirements, basic knowledge of sets, functions and equivalence relations is expected.

On completion of the course, students will:

• arrow_rightbe familiar with concepts and techniques in group theory;
• arrow_rightbe able to identify real-life problems that can be modeled with abstract algebra;
• arrow_rightbe familiar with mathematical formality and rigor.

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 and uses in cryptography and error correcting codes
   6. Arithmetic with congruence classes
2. Group theory (5 weeks)
   1. Definition of a group and real-life examples such as Rubik's cube movements
   2. Properties of group elements
   3. Subgroups
   4. Cyclic groups
   5. Homomorphisms
   6. Isomorphisms
   7. Finite permutation groups and uses in physics, music, and entertainment
   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 for modeling with polynomials
   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%)