Math 631 — Algebra

Fall 2002

Instructor: Yuri Berest

Time: WF 1:25-2:40

Room: Malott 224

Office Hours (tentative): WF 3:00-4:30 pm and/or by appointment

This is a basic graduate course in Algebra. I plan to cover a standard range of topics (groups, fields, rings and modules) following mostly S. Lang's Algebra. There will be weekly homework assignments (40%), a (take home) midterm (30%) and a (take home) final (30%).

Topics to be covered in Course include:

I. Group Theory:

Basic definitions (subgroups, normal subgroups, factor groups, etc.), homomorphism/isomorphism theorems, actions of groups on sets, finite groups, p-groups, Sylow theorems, solvable groups, composition series, Jordan-Holder theorem, examples.

II. Field Theory:

Fields, field extensions, degree, splitting fields, algebraically closed fields, separability, Galois Theory, applications of Galois Theory (solvability of equations in radicals), finite fields and finite geometries, examples.

III. Ring Theory:

Basic definitions (rings and algebras), some general techniques: homomorphism/isomorphism theorems, localization, filtering, completions. Modules: free, injective, projective, (very) basic homological algebra: exact sequences, tensor products and Tor's, Hom's and Ext's. Examples.