MATH 631: Algebra (Fall 2006)

Instructor: Shankar Sen

Meeting Time & Room

Prerequisite: Assumes familiarity with the material of standard undergraduate course in abstract algebra. (This includes basic definitions and properties of groups, rings, modules and their homomorphisms; sub- and quotient structures; isomorphism theorems; integral domains and their fraction fields. Very little of this, if any, will be covered in 631.)

MATH 631-632 are the core algebra courses in the mathematics graduate program. 631 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. Optional topic: introduction to affine algebraic geometry. (Basic representation theory of finite groups, introductory homological algebra, and Galois theory are covered in 632.)

In slightly more detail the topics covered will (probably) include:

Group Theory: group action on sets, p-groups and the Sylow theorems, composition series and the Jordan-Holder theorem for groups.

Rings, Fields, Modules: maximal and prime ideals, the Chinese remainder theorem, principal ideal domains and unique factorization domains, Noetherian rings, polynomial rings and the Hilbert basis theorem, basics of field extensions, structure of finitely generated modules over principal ideal domains.

Multilinear Algebra: tensor products of modules, the tensor and exterior algebras of a module.

Text: The main text for Math 631 is Dummit & Foote, Abstract Algebra, 3rd edition, 2004.

Last modified: April 25, 2006