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 & |
|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.
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