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
- Isomorphism Theorems, composition series, Jordan-Holder, simple groups.
- Sylow Theorems, class equation, applications.
- Solvable and nilpotent groups
- Finite products, semi-direct products, group extensions
- Categories, products, coproducts.
- Products and coproducts of groups, free groups, generators and relations.
- Geometric applications, isometry groups, fundamental groups.
B. Ring Theory
- Definitions and basic examples
- Homemorphisms, quotients, ideals.
- Localization, Chinese remainder theorem.
- The spectrum of a commutative ring.
- Polynomial rings, power series.
- Discrete valuation rings, Dedekind rings.
C. Module Theory.
- Homomorphisms, quotients, direct sums and products.
- Noetherian rings and modules, Hilbert basis theorem, power series,
Artinian modules.
- Vector spaces, dual spaces, determinants.
- Projective, injective and flat modules.
- Limits and colimits.
- Finitely generated modules over a PID.
- Jordan and rational canoncial forms.
- Local rings, Nakayama's Lemma.
D. Linear algebra
- Simplicity and semi-simplicity.
- Density Theorem, Wedderburn Theorem
- Matrices and bilnear forms. Symmetric and hermitian forms.
- Structure of bilinear forms
- Spectral Theorem
- Tensor product, symmetric product, alternating product.
- Representations of groups, characters
E. Field theory and Galois Theory
- Definitions, examples, finite and algebraic extensions.
- Algebraic closure, splitting fields, normal extensions.
- Separable and inseparable extensions.
- Galois extenions, Galois groups, fundamental theorem of Galois
theory.
- Examples (finite fields, cyclotomic fields, Kummer extensions,
computation of Galois groups).
- Norms and traces.
- Cyclic, solvable, radical, abelian extensions. Applications.
F. Homological algebra
- Complexes, resolutions, derived functors.
- (Co)homological functors.
- Spectral sequences and applications.
- 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.