Cornell Math - MATH 631, Fall 2003

MATH 631
Algebra
 (Fall 2003)

Instructor: Steve Chase

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 has not yet been selected. In any case, several algebra texts at both the undergraduate and graduate levels will be put on reserve for the course.

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 formulae 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. Co-maximal (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. Introduction to Algebraic Geometry

  1. Algebraic sets and varieties.
  2. Hilbert's Nullstellensatz.
  3. Nilpotent elements and radical of commutative ring and ideal.

IV. Multilinear Algebra

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

Additional topics related to the above will be covered as time permits.