# Math 533: Abstract Algebra I

### Fall 2016

Mondays and Wednesdays 12:30-1:50pm, One Drexel Plaza 007

Course Description: Introduction to rings and modules, structure theorem for modules over a principal ideal domain, representation theory, Artin-Wedderburn theorem, tensor products, symmetric and exterior algebras, Schur functions, Schur-Weyl duality.
Prerequisites: at least one semester of abstract algebra.
Office Hours: Korman 275, Tuesday 2:30-4pm, Thursday 1-2:30pm.
Problem Session: Korman Lounge, Tuesday 2-4pm.
Required text: Abstract Algebra, 3rd Edition, David S. Dummit and Richard M. Foote.
Other references:
• The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd Edition, Bruce E. Sagan.
Homework Policy: You may consult each other and the textbooks above. List all people and sources who aided you and whom you aided, and write up the solutions independently, in your own language. Many exercises will be taken from Dummit and Foote. Solutions to these are available online. Please do not consult these until after you have turned in your homework.
• 100% homework
• Important University Policies:

Disability Resources

Course Drop Policy

Code of Conduct

## Syllabus

Week 1: Sep 19 (Mon), Sep 21 (Wed)
Definitions and examples of rings and ideals, isomorphism theorems
Read 7.1--7.4 in Dummit and Foote
Problem Set 1, Due Wednesday, Sep 28.

Week 2: Sep 26 (Mon), Sep 28 (Wed)
Prime ideals and maximal ideals, definitions and examples of modules
Read 7.4, 10.1--10.2 in Dummit and Foote
Problem Set 2, Due Wednesday, Oct 5.

Week 3: Oct 03, Oct 05
Module isomorphism theorems, products, coproducts, bases, rank
Problem Set 3, Due Wednesday, Oct 12.

Week 4: Oct 12
Chinese Remainder Theorem, principal ideal domains
Problem Set 4, Due Wednesday, Oct 19.

Week 5: Oct 17, Oct 19
Structure theorem for modules over a principal ideal domain
The Wikipedia page on Smith normal form is close to the proof given in class.
Problem Set 5, Due Wednesday, Oct 26.

Week 6: Oct 24, Oct 26
Rational canonical form, Jordan canonical form, Representation Theory
Problem Set 6, Due Wednesday, Nov 2.

Week 7: Oct 31, Nov 02
Representation Theory, Character Theory
Problem Set 7, Due Wednesday, Nov 9.

Week 8: Nov 07, Nov 09
Character Theory
Problem Set 8, Due Wednesday, Nov 16.

Week 9: Nov 14, Nov 16
Artin-Wedderburn Theorem, Determinants
Problem Set 9, Due Wednesday, Nov 30.

Week 10: Nov 21
Tensor Products

Week 11: Nov 28, Nov 30
Tensor products, tensor algebra, exterior algebra

We will cover some subset of the following topics, to be refined as the course progresses:

Definitions and examples of rings and ideals, isomorphism theorems
Definitions and examples of modules, isomorphism theorems
Polynomial rings