Algebra studies the structure of sets with operations, such as integers with addition and multiplication, or vector spaces with linear maps. The abstract point of view, based on an axiomatic approach, reveals many deep ideas behind seemingly innocent structures—such as the arithmetic of counting numbers—and serves as an elegant organizing tool for the vast universe of modern algebra. Generations of brilliant minds have crystallized these ideas in the ideas in the concept of groups, rings, fields, modules and their quotient structures and homomorphisms—the topics of MATH 335 & 435. The main goal in MATH 335 is the study of groups and rings. Another goal is to make the students immerse in communicating mathematical thoughts (proofs, examples, counterexamples) in a written form.
Evaluation of Students
The students will be evaluated based on frequent homework assignments, quizzes, midterm and final exam.
1. Groups (10 weeks):
Definition and basic properties of groups, examples such as Zn, dihedral groups, matrix groups. Special emphasis on symmetric groups (cycle notation, generators, transpositions, even/odd permutations) to emphasize the non-abelian structure. Subgroups (examples such as normalizers, centralizers, subgroups generated by subsets). Cyclic groups (classification of subgroups of cyclic groups). Homomorphisms, isomorphisms, and normal subgroups (images and kernels). Left and right cosets, Lagrange's theorem, factor/quotient groups, isomorphism theorems. Time permitting the statement of the Fundamental Theorem of Finitely Generated Abelian Groups and applications.
2. Rings (5 weeks):
Definition of rings, unit, zero-divisor, division ring, integral domain, field; basic properties; examples: integers, polynomial rings, power series rings. Ring homomophisms and isomorphisms, subrings, images and kernels, definition of ideals and quotient rings, one- and two-sided ideals, isomorphism theorems, properties of ideals. PIDs, greatest common divisor. UFDs, division algorithm, irreducibility of polynomials.
Textbooks & Software
J. Gallian, Contemporary Abstract Algebra, Houghton Mifflin
N. Lauritzen, Concrete Abstract Algebra, Cambridge University Press.
Submitted by: Matthias Beck, Joseph Gubeladze and Serkan Hosten
Date: June 2011