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Dr. Matt Insall will teach the course Modern Algebra II/Introduction to Ring Theory this semester. It is two courses in one. Modern Algebra II is a continuation of Modern Algebra I (MATH 5105); its course number is MATH 5106. Introduction to Ring Theory is a graduate course continuing in the same vein, but specialized to algebraic structures in the same kind of arithmetic used in the integers or matrix theory or linear algebra.
This course is designed for students interested in going further in computational science related to the following applied areas of mathematics: cryptography, computer science, natural language computation, applied mathematical logic and computer engineering.
Some of these connections include:
1. Quantum logic is an area of applied mathematical logic that generalizes, for use with reasoning in quantum systems, the theory of Boolean logic, and the algebraic treatment of Boolean logic related to Boolean circuits and Boolean circuit design involves Boolean algebras and Boolean rings.
2. Classical algebraic number theory is fundamental to cryptography, and classical algebraic number theory is rooted in the theory of the ring of integers and extends to rings such as the Gaussian integers and the field of algebraic numbers.
3. Ring theory and logic help provide verification tools for computational optimization and optimized computation in the interplay between physical and digital systems that is required for increased certainty about the security, safety and reliability of engineering systems.
4. Ring theory provides the foundational information about Persistent Homology which is used in topological data analysis.
5. Ring theory provides the foundation for the theory of Chip Firing modeled via games on graphs using Divisors and Sandpile Arithmetic.
Here are the course descriptions:
MATH 5106 Modern Algebra II (LEC 3.0)
This course is a continuation of MATH 5105. Rings and fields are discussed. Euclidean domains, principal ideal domains, unique factorization domains, vector spaces, finite fields and field extensions are studied. Prerequisite: MATH 5105 (or instructor approval).
MATH 6106 Introduction to Ring Theory (LEC 3.0)
Properties of rings with an emphasis on commutative rings. Ideals, factor rings, ring homomorphisms, polynomial rings; factorization, divisibility, and irreducibility. Introduction to extension fields and Galois theory. Applications may be chosen based on the interests of the students. Prerequisite: MATH 5105 (or instructor approval).
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