McGill University

Department of Mathematics & Statistics

Higher Algebra II


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.