## McGill University

# Department of Mathematics & Statistics

# Higher Algebra II

# 189-571B

## Detailed Syllabus

Whereas in the first semester we focussed mainly on group theory, representation theory, and Galois theory, the unifying concept for the second
semester will be the theory of rings and modules.

** 1. Monday January 8 to Friday January 12**

**Text**: Basic Algebra, Chapters I and VIII.

Overview of the course. Basic notions of ring theory. Definitions,
some key examples.

Assignment 1 (Due Wednesday, January 24.)

** 2. Monday January 15 to Friday January 19 **

**Text**: Basic Algebra, Chapter VIII.

Commutative rings and their modules.

** 3. Monday January 22 to Friday January 26 **

**Text**: Basic Algebra, Chapter VIII.

Commutative rings and their modules, cont'd.

** 4. Monday January 29 to Friday February 2 **

**Text**: Basic Algebra Chapter VIII.

Dedekind rings.

Assignment 2 (Due Wednesday, February 7.)

** 5. Monday February 5 to Friday February 9 **

**Text**: Basic Algebra, Chapter X.

Dedekind rings, cont's. Modules over non-commutative rings.

** 6. Monday February 12 to Friday February 16 **

**Text**: Advanced Algebra, Chapter II.

Artinian rings. Finite-dimensional semisimple modules over
*F* algebras.

Assignment 3 (Due Wednesday, February 21.)

** 7. Monday February 19 to Friday February 23 **

**Text**: Advanced Algebra, Chapter II.

Wedderburn-Artin ring theory, cont'd.

**8. Monday February 26 to Monday March 2**.

**Text**: Advanced Algebra, Chapter II.

Classification of central simple algebras and division algebras, cont'd.

Assignment 4 (Due Wednesday, March 14.)

** Monday March 5 to Friday March 9 **

Study Break

** 9. Monday March 12 to Friday March 16 **

**Text**: Advanced Algebra, Chapter II.

The *Midterm* will be held on Monday, March 12, in class.

On Wednessday we covered the last important topics in Chapter II,
namely the theorems of Frobenius and Wedderburn on the classification of
division algebras over the reals and over finite fields, respectively.

** 10. Monday March 19 to Friday March 23 **

*On week 10, we will have an extra class on Friday, to make up for the fact
that classes will be cancelled in week 11. *

Advanced Algebra, Chapter III

The Brauer Group.

** Important scheduling change:**
*On week 11, we will not have class on Monday and Wednesday. We will not
meet on
Friday either, because of the Easter holiday.*

** 11. Monday March 26 to Friday March 30 **

Advanced Algebra, Chapter III, cont'd.

Assignment 5 (Due Wednesday, April 4.)

** Important scheduling change:**
*On week 12, we will have an additional class
on Friday to make up for one of the classes that was missed in the previous
week.*

** 12. Monday April 2 to Friday April 6 **

**Text**: Advanced Algebra, Chapter III.

** 13. Monday April 9 to Friday April 13 **

Revision of the material covered in class.
Here is a Practice Final Exam to aid you in your studies.

The final exam will be held in class on Monday, April 23, from 9:00 AM to 12:00 PM.