MATH 6310: Algebra (fall 2008)

Instructor: R. Keith Dennis

Prerequisites: The content of a solid undergraduate course in abstract algebra, including basic definitions and properties of groups, rings, modules, and homomorphisms of such; sub- and quotient structures; isomorphism theorems; integral domains and their fraction fields. Very little, if any, of this material will be reviewed during the course.

The following is a general outline of MATH 6310:

I. Group Theory

  1. Composition series and Jordan-Holder theorem in context of groups with operators; simple groups and modules; solvable groups.
  2. Group actions on sets and groups; orbit formulas for action of a group on a finite set; class equation.
  3. p-Groups and Sylow theorems.
  4. Free groups; generators and relations.

II. Rings, Fields, Modules

  1. Maximal and prime ideals; existence of maximal left ideals and relation to Zorn's Lemma.
  2. Comaximal (relatively prime) ideals and general Chinese Remainder Theorem.
  3. Noetherian rings.
  4. Principal ideal domains and unique factorization domains.
  5. Polynomial rings; Hilbert's Basis Theorem; Gauss's Lemma.
  6. Finite, algebraic, and primitive field extensions; degree formula for finite field extensions.
  7. Free modules; structure of finitely generated modules over principal ideal domains.

III. Multilinear Algebra

  1. Tensor product of modules.
  2. Tensor algebra of a bimodule.
  3. Exterior algebra of a module over a commutative ring.

IV. Additional topics will be covered as time permits.

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.