# MATH 6310: Algebra (Fall 2010)

**Instructor:**
Ken Brown

**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 Jordan-Hölder theorem in context of groups with operators; simple groups and modules; solvable 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

MATH 6310 is the first semester of a two-semester basic graduate algebra sequence. The main topics to be covered in the second semester, MATH 6320, are Galois theory, representation theory of groups and associative algebras, and an introduction to homological algebra.