## 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 III.

The Brauer group.

The *Midterm* will be held this week, on Wednesday, February 28,
at 8:30 AM, in room 920.

** Monday March 5 to Friday March 9 **

Study Break

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

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

The Brauer group, cont'd.

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

Advanced Algebra, Chapter VIII

Background for algebraic geometry.

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

Advanced Algebra, Chapter VIII.

Background for algebraic geometry, cont'd.

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

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

The number theory of algebraic curves.

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

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

The number theory of algebraic curves, cont'd.

Practice Final