Math 602

Math 602

ALGEBRA

This is the first half of a year long course in graduate algebra, with emphasis on the ways in which category theory, algebra, and geometry interact and complement each other. In the 602/603 sequence we will cover the following topics:

A. Group Theory
  1. Isomorphism Theorems, composition series, Jordan-Holder, simple groups.
  2. Sylow Theorems, class equation, applications.
  3. Solvable and nilpotent groups
  4. Finite products, semi-direct products, group extensions
  5. Categories, products, coproducts.
  6. Products and coproducts of groups, free groups, generators and relations.
  7. Geometric applications, isometry groups, fundamental groups.

B. Ring Theory
  1. Definitions and basic examples
  2. Homemorphisms, quotients, ideals.
  3. Localization, Chinese remainder theorem.
  4. The spectrum of a commutative ring.
  5. Polynomial rings, power series.
  6. Discrete valuation rings, Dedekind rings.

C. Module Theory.
  1. Homomorphisms, quotients, direct sums and products.
  2. Noetherian rings and modules, Hilbert basis theorem, power series, Artinian modules.
  3. Vector spaces, dual spaces, determinants.
  4. Projective, injective and flat modules.
  5. Limits and colimits.
  6. Finitely generated modules over a PID.
  7. Jordan and rational canoncial forms.
  8. Local rings, Nakayama's Lemma.

D. Linear algebra
  1. Simplicity and semi-simplicity.
  2. Density Theorem, Wedderburn Theorem
  3. Matrices and bilnear forms. Symmetric and hermitian forms.
  4. Structure of bilinear forms
  5. Spectral Theorem
  6. Tensor product, symmetric product, alternating product.
  7. Representations of groups, characters

E. Field theory and Galois Theory
  1. Definitions, examples, finite and algebraic extensions.
  2. Algebraic closure, splitting fields, normal extensions.
  3. Separable and inseparable extensions.
  4. Galois extenions, Galois groups, fundamental theorem of Galois theory.
  5. Examples (finite fields, cyclotomic fields, Kummer extensions, computation of Galois groups).
  6. Norms and traces.
  7. Cyclic, solvable, radical, abelian extensions. Applications.

F. Homological algebra
  1. Complexes, resolutions, derived functors.
  2. (Co)homological functors.
  3. Spectral sequences and applications.
  4. Koszul complex, syzygies, derivations.

A basic goal of the course is to use the abstract theory to develop an intuition in concrete examples and learn to understand and produce sound mathematical arguments.