MATH 6310 - Algebra

Marcelo Aguiar, fall 2015.

Prerequisites: The content of a solid undergraduate course in abstract algebra, comparable to MATH 4340. Students should know the basic definitions and properties of groups, rings, modules, and homomorphisms; substructures and quotient structures; isomorphism theorems; integral domains and their fraction fields. Very little of this material will be reviewed during the course.

Topics

I. Group theory

Composition series and the Jordan-Hölder theorem in the context of groups with operators; simple groups and modules; solvable and nilpotent groups
Group actions
p-Groups and Sylow theorems
Free groups; generators and relations

II. Rings, fields, modules

Maximal and prime ideals
Comaximal ideals and Chinese Remainder Theorem
Noetherian rings
Principal ideal domains and unique factorization domains
Polynomial rings; Hilbert's Basis Theorem; Gauss's Lemma
Finite, algebraic, and primitive field extensions
Presentations of modules; structure of finitely-generated modules over principal ideal domains

III. Introduction to algebraic geometry

Algebraic sets and varieties
Hilbert's Nullstellensatz
Nilpotent elements and radical

IV. Multilinear Algebra

Tensor product of modules
Tensor algebra of a bimodule
Exterior algebra of a module over a commutative ring

The main text is David S. Dummit & Richard M. Foote, Abstract Algebra, 3rd ed., John Wiley & Sons, 2004 (ISBN 0-471-43334-9).

Additional references include:

I. M. Isaacs, Algebra, a graduate course, 1994;
T. W. Hungerford, Algebra, 1974;
N. Jacobson, Basic algebra, two volumes, 2nd edition, 1985–1989;
S. Lang, Algebra, 3rd edition, 2002.