Homework Assignments (Fall 2010)
This is a standard course for beginning graduate students. It covers:
| Group Theory | 10 lectures | Sept.1-Oct.4 | Midterm |
| Basic Ring & Module Theory | 9 lectures | Oct.7-Nov.1 | |
| Modules over a PID | 3 lectures | Nov.4-11 | |
| Bilinear Forms | 3 lectures | Nov. 15-18,29 | |
| Artin-Wedderburn & Maschke theorems | 4 lectures | Dec.2-13 | Final Exam |
Group Theory: Basic concepts, isomorphism theorems,
normal subgroups, Sylow theorems, direct products and free products of groups.
Groups acting on sets: orbits, cosets, stabilizers.
Alternating/Symmetric groups.
Basic Ring Theory: Fields, Principal Ideal Domains (PIDs),
matrix rings, division algebras, field of fractions.
Modules over a PID: Fundamental Theorem for abelian groups,
application to linear algebra: rational and Jordan canonical form.
Bilinear Forms: Alternating and symmetric forms, determinants.
Spectral theorem for normal matrices,
classification over R and C.
(
Class supplement provided)
Modules: Artinian and Noetherian modules.
Krull-Schmidt Theorem for modules of finite length.
Simple modules and Schur's Lemma, semisimple modules.
(from Basic Algebra II)
Finite-dimensional algebras: Simple and semisimple
algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.
(
Class supplement provided)
Last updated: November 1, 2010