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 435 is the study of rings and modules, with applications in computational algebra.
Evaluation of Students
The students will be evaluated based on frequent homework assignments, midterm and final exam.
1. Groups (3 - 4 weeks):
Quick review of fundamentals of group theory emphasizing quotient groups. Definition of group actions and examples, orbits, conjugation action, conjugacy class, conjugacy classes in the symmetric group. The statement of Sylow's Theorem and its simple applications.
2. Modules and Vector Spaces (4 weeks):
Definition of modules, abelian groups as Z-modules, vector spaces as F-modules, vector spaces with linear transformations as F[x]-modules, submodules module homomorphisms, quotient modules, isomorphism theorems, direct sums, free modules. The statement of the main theorem of finitely generated modules over PIDs.
3. Fields and Field Extensions (4 weeks):
Maximal and prime ideals. Characteristic of a field, field extension, degree of an extension, computing in a finite extension, algebraic extension, minimal polynomial, roots of unity. Splitting fields. Finite fields.
4. Miscellaneous topics (3 - 4 weeks):
Basics of a classic or well-established topic that could be built upon what has been done in the two modern algebra courses. These include the rudiments of Galois theory, Groebner bases and basic computational algebraic geometry, algebraic coding theory, algebraic number theory, etc.
Textbooks & Software
N. Lauritzen, Concrete Abstract Algebra, Cambridge University Press.
Submitted by: Matthias Beck, Joseph Gubeladze and Serkan Hosten
Date: October 29, 2006