MATH 4GR3, Winter 2019
GROUPS AND RINGS
The goal of this course is to delve deeper in the study of the algebraic structures introduced in the course MATH 3GR3, namely groups, rings, and fields. The first third of the course will consider the structure of finite groups, starting first with abelian groups. We will prove the fundamental theorem of finite abelian groups, which shows that up to isomorphism, every finite abelian group is equal to a cartesian produce of cyclic groups. In the non-abelian setting, things are much more complicated, but by developing the notion of a group action, we will be able to prove the Sylow theorems, and use them to establish important and useful structural properties of finite groups. The second part of the course will explore the notions of primeness in ring theory in the setting of integral domains. We will show that the familiar property of the natural numbers, that every natural number can be uniquely factored (up to reordering) as a product of prime numbers, has a natural extension to a much wider class of arithmetic structures. The last part of the course will investigate fields and field extensions. If time permits, algebraically closed fields will be studied. These are fields with the property that any polynomial over them will have at least one root in them. The prerequisite for this course is MATH 3GR3. Students will be evaluated using a set of graded assignments, midterm tests, and a final exam.
INSTRUCTOR: M. Valeriote
Further topics in group theory and ring theory. Topics include: direct products, Fundamental Theorem of Finite Abelian Groups, Sylow Theorems, free groups, group presentations, fields and integral domains, special integral domains (Euclidean, principal ideal, unique factorization), fields of fractions of integral domains, polynomial rings in many variables, and additional topics at the discretion of the instructor (e.g., Groebner bases, algebraic coding theory.)
Three lectures; one term
Prerequisite(s): MATH 3E03 or 3GR3
PLEASE REFER TO MOSAIC FOR THE MOST UP-TO-DATE INFORMATION ON TIMES AND ROOMS