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
- Group actions and the class equation
- The Sylow theorems
- Products and semi-direct products
- Composition series, the Jordan-Hoelder theorem
- The Krull-Remak-Schmidt theorem
- Coproducts and free groups
- Generators and relations
B. Ring Theory
- Definitions and basic examples
- Homorphisms, quotients, ideals
- Product rings
- Fractions and localization
- Maximal and prime ideals
- Factoring integers
- Unique factorization domains
- Gauss's lemma
C. Module Theory.
- Definitions and basic examples
- Operations on modules
- Free modules
- Generators and relations
- Noetherian rings
- Modules over a PID
D. Field theory.
- Examples
- Algebraic and transcendental elements
- Field extensions
- Finite fields
- Primitive elements
- The Fundamental Theorem of Algebra
E. Galois theory.
- Splitting fields
- Isomorphisms of field extensions
- Fixed fields
- Galois extensions
- Main theorem
- 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.