# Math 533: Abstract Algebra I

### Winter 2024

Mondays and Wednesdays 12:30-1:50pm, Curtis Hall, Room 454

**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.

**Level:** graduate.

**Office Hours:** Korman 241, Mondays 2pm-3pm, Tuesdays 4:30-6pm.

**Problem Session:** Korman 241, Tuesdays 3-6pm.

**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.

**Grading policy:**

100% homework
**
Please visit this site (https://www.math.drexel.edu/~jblasiak/AlgebraMath533Winter2024.html) for new information. Updates to the syllabus, reading assignments, and homeworks will be posted here as the course progresses.
**

## Syllabus

Week 1: Jan 08 (Mon), Jan 10 (Wed)

Definitions and examples of rings and ideals, isomorphism theorems

Read 7.1--7.3 in Dummit and Foote

Problem Set 1, Due Wednesday, Jan 17.

Week 2: Jan 17 (Wed)

Maximal ideals, definitions and examples of modules

Read 7.4, 10.1--10.2 in Dummit and Foote

Problem Set 2, Due Wednesday, Jan 24.

Week 3: Jan 22, Jan 24

Prime ideals, module isomorphism theorems, products, coproducts

Read 10.2--10.3

Problem Set 3, Due Wednesday, Jan 31.

Week 4: Jan 29, Jan 31

Bases, rank, Chinese Remainder Theorem, principal ideal domains

Read 7.6, skim 8.1--8.3

Problem Set 4, Due Wednesday, February 7.

Week 5: Feb 05, Feb 07

Structure theorem for modules over a principal ideal domain

The Wikipedia page on Smith normal form is close to the proof given in class.

Read 12.1--12.2

Problem Set 5, Due Wednesday, February 14.

Week 6: Feb 12, Feb 14

Rational canonical form, Jordan canonical form

Read 12.2--12.3

Problem Set 6, Due Wednesday, February 21.

Week 7: Feb 19, Feb 21

Representation Theory

Read 18.1

Problem Set 7, Due Wednesday, February 28.

Week 8: Feb 26, Feb 28

Character Theory

Read 18.1, 18.3

Problem Set 8, Due Wednesday, March 6.

Week 9: Mar 04, Mar 06

Character Theory

Read 18.3, 22.1

Problem Set 9, Due Wednesday, March 13.

Week 10: Mar 11, Mar 13

Artin-Wedderburn Theorem

Read 18.2

**Outcomes:**
The students should have an understanding of the basic objects of abstract algebra including groups, rings, and modules. They should also gain an understanding of homomorphisms, direct sums and products, and tensor products. They should have a knowledge of the basic theorems in this area including isomorphism theorems and universal properties. They should be familiar with representation theory and the structure theorem for modules over a principal ideal domain.