# Math 533: Abstract Algebra I

### Winter 2018

Tuesdays and Thursdays 3:30-4:50pm, One Drexel Plaza 137

**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:** Academic Building 322, Wednesday 1:30pm-3pm.

**MRC Hours:** Wednesday 3pm-5pm, Thursday 6pm-7pm.

**Problem Session:** Academic Building 3rd Floor Lounge, Wednesday 2pm-5pm.

**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

## Syllabus

Week 1: Jan 09 (Tue), Jan 11 (Thu)

Definitions and examples of rings and ideals, isomorphism theorems

Read 7.1--7.3 in Dummit and Foote

Problem Set 1, Due Thursday, Jan 18.

Week 2: Jan 16 (Tue), Jan 18 (Thu)

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 Thursday, Jan 25.

Week 3: Jan 23, Jan 25

Module isomorphism theorems, products, coproducts, bases, rank

Read 10.2--10.3

Problem Set 3, Due Thursday, February 1.

Problem 9 from Problem Set 2 was moved to Problem 5 on this set.

Week 4: Jan 30, Feb 01

Chinese Remainder Theorem, principal ideal domains

Read 7.6, skim 8.1--8.3

Problem Set 4, Due Thursday, February 8.

Week 5: Feb 06, Feb 08

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 Thursday, February 15.

Week 6: Feb 13, Feb 15

Rational canonical form, Jordan canonical form

Read 12.2--12.3

Problem Set 6, Due Thursday, February 22.

Week 7: Feb 20, Feb 22

Representation Theory, Character Theory

Read 18.1, 18.3

Problem Set 7, Due Thursday, March 1.

Week 8: Feb 27, Mar 01

Character Theory

Read 18.3, 19.1

Problem Set 8, Due Thursday, March 8.

Week 9: Mar 06, Mar 08

Character Theory
Read 18.3, 19.1

Problem Set 9, Due Thursday, March 15.

Week 10: Mar 13, Mar 15

Artin-Wedderburn Theorem, Tensor Products

Read 18.2, 10.4

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

Graded rings and modules

Jordan-Hölder Theorem

Products, coproducts, universal properties

Chinese Remainder Theorem

Structure theorem for modules over a principal ideal domain

Artin-Wedderburn Theorem

Introduction to Representation Theory: irreducible representations, Maschke's Theorem, Schur's Lemma, character table, orthogonality relations

Tensor products, symmetric and exterior algebra, determinants

Representations of the symmetric group

**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.