Math 503 Sprin 2015

Math 503

ALGEBRA

This is the second half of a year long course in algebra. In the Math 503 we will cover some advanced group theory, basic ring theory and commutative algebra, field extensions, and Galois theory.

A. Group Theory
  1. Group actions and the class equation
  2. The Sylow theorems
  3. Products and semi-direct products
  4. Composition series, the Jordan-Hoelder theorem
  5. The Krull-Remak-Schmidt theorem
  6. Coproducts and free groups
  7. Generators and relations

B. Ring Theory
  1. Definitions and basic examples
  2. Homorphisms, quotients, ideals
  3. Product rings
  4. Fractions and localization
  5. Maximal and prime ideals
  6. Factoring integers
  7. Unique factorization domains
  8. Gauss's lemma

C. Module Theory.
  1. Definitions and basic examples
  2. Operations on modules
  3. Free modules
  4. Generators and relations
  5. Noetherian rings
  6. Modules over a PID

D. Field theory.
  1. Examples
  2. Algebraic and transcendental elements
  3. Field extensions
  4. Finite fields
  5. Primitive elements
  6. The Fundamental Theorem of Algebra

E. Galois theory.
  1. Splitting fields
  2. Isomorphisms of field extensions
  3. Fixed fields
  4. Galois extensions
  5. Main theorem
  6. Examples

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.