Hecke Algebras and Orthogonal Polynomials

Research Focus Group

Department of Mathematics

University of California Davis

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Summer/Fall 2004

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 practice problems, or in pdf 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/ suggested texts. 

Course Description for 250ABC: Group and rings. Sylow theorems, abelian groups, Jordan-Holder theorem. Rings, unique factorization. Algebras, and modules.  Fields and vector spaces over fields. Field extensions. Commutative rings. Representation theory and its applications.
TOPICAL OUTLINE -- sept 8, 9 (probably):
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 (probably):
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 category theory.

250ABC Textbook: Abstract algebra, Dummit and Foote.
Additional reference books:
    Artin, Algebra.
    Rose, A course on group theory.
    Dixon, Problems in group theory.
    Hall, The theory of groups.