Hecke Algebras and Orthogonal
Research Focus Group
Review Session in algebra for the
Prelim Exam (2004)
for graduate students.
Sept 8-10, Sept 13 (Wednesday-Friday, Monday)
We will have 2 hours of lecture/review in the mornings
from 10 AM - noon, in 693 Kerr,
and problem sessions in the afternoon, from 3 PM - 5 PM.
Check back here for details.
Click here for a .ps version of some
which have just been updated, with a new section on category theory
(and a typo fixed).
Please do as many as you can by the afternoon sessions.
There YOU will present solutions and have a chance to ask questions.
Solutions will be posted here, but only AFTER you've had some time
to solve these on your own.
Thanks everyone for participating! Here are some solutions to
section 1 (groups)
section 5 (groups)
section 4 (modules)
section 3 (fields and galois theory).
More info on the exam:
The algebra portion
of the exam will be based on the syllabus for Math 250A,B,C. A
short course description and more detailed outline are below, along w/
Description for 250ABC: Group and rings. Sylow theorems, abelian
theorem. Rings, unique factorization. Algebras, and modules. Fields and vector spaces over fields.
Field extensions. Commutative rings. Representation theory and its
TOPICAL OUTLINE -- sept 8, 9
Group Theory: groups as symmetries,
homomorphisms, subgroups and quotient groups, group actions, abelian groups.
Basic concepts including group actions, Sylow's theorems, classifications of
finite abelian groups and groups of small order, the Jordan-Holder theorem.
Introduction to rings, ring
homomorphisms and ideals, field of fractions of a commutative domain; polynomial
rings; factorial rings; principal ideal domains and Euclidean domains;
polynomial extensions of factorial domains. Unique factorization. Free
groups, group representations.
TOPICAL OUTLINE -- sept 10, 13
Modules and vector spaces. Dual
vector spaces, multilinear functions, determinants, bilinear forms, tensor
products and tensor algebras, the
classification of finitely generated abelian groups and endomorphisms of finite-dimensional
vector spaces, field theory and Galois theory. Time permitting also
250ABC Textbook: Abstract algebra,
Dummit and Foote.
Additional reference books:
Rose, A course on
in group theory.
Hall, The theory