# Algebra and its Applications

 COURSE DESCRIPTION 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. Faculty Fall 2019 Luis NÚÑEZ BETANCOURT
 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 Basic number theory (4 weeks)Postulates for the integers Mathematical induction Divisibility Prime factors and greatest common divisors Congruence of integers and uses in cryptography and error correcting codes Arithmetic with congruence classes Group theory (5 weeks)Definition of a group and real-life examples such as Rubik's cube movements Properties of group elements Subgroups Cyclic groups Homomorphisms Isomorphisms Finite permutation groups and uses in physics, music, and entertainment Cayley's Theorem Cosets of a subgroup Normal subgroups Quotient groups Direct sums Rings and Fields (4 weeks)Definition of a ring Integral domains and fields The field of quotients of an integral domain  Ideals and quotient rings for modeling with polynomials Ring homomorphisms Bibliography Gilbert, Linda; Gilbert, Jimmie (2015). Elements of Modern Algebra, Eighth Edition, Cengage. Beachy, John A.; Blair, William D. (2005). Abstract Algebra, Third Edition, Waveland Press. 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%)