# 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.

**Level:** graduate.

**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.

**Grading policy:**

100% homework

## 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

Read 10.2--10.3

Problem Set 3, Due Wednesday, Oct 12.

Week 4: Oct 12

Chinese Remainder Theorem, principal ideal domains

Read 7.6, skim 8.1--8.3

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.

Read 12.1--12.3

Problem Set 5, Due Wednesday, Oct 26.

Week 6: Oct 24, Oct 26

Rational canonical form, Jordan canonical form, Representation Theory

Read 12.1--12.2, 18.1

Problem Set 6, Due Wednesday, Nov 2.

Week 7: Oct 31, Nov 02

Representation Theory, Character Theory

Read 18.1, 18.3

Problem Set 7, Due Wednesday, Nov 9.

Week 8: Nov 07, Nov 09

Character Theory

Read 18.3, 19.1

Problem Set 8, Due Wednesday, Nov 16.

Week 9: Nov 14, Nov 16

Artin-Wedderburn Theorem, Determinants

Read 18.2, 11.4

Problem Set 9, Due Wednesday, Nov 30.

Week 10: Nov 21

Tensor Products

Read 10.4

Week 11: Nov 28, Nov 30

Tensor products, tensor algebra, exterior algebra

Read 10.4, 11.5

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.