MATH 631: Algebra (Fall 2005)

Instructor: R. Keith Dennis

Meeting Time & Room

This course is the first semester of a two-semester basic graduate algebra sequence. (The main topics to be covered in the second semester, MATH 632, are Galois theory, representation theory of groups and associative algebras, and an introduction to homological algebra.)

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.

Text: A text for MATH 631 is Dummit & Foote, Abstract Algebra, 3rd edition, 2004.

Syllabus:

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.

Last modified: August 19, 2005