McGill University

Department of Mathematics & Statistics

Higher Algebra II

189-571B

Detailed Syllabus



The first semester focused mainly on group theory, representation theory, Galois theory, and the rudiments of the language of schemes. The unifying concept for the second semester will be a deeper incursion into the theory of rings and modules, including (a) Dedekind rings and unique factorisation into prime ideals, forming the backdrop for algebraic number theory, (b) polynomial rings and their quotients, as a prelude to algebraic geometry, and (c) the theory of semisimple and simple algebras over a field, and the Brauer group.


1. Monday January 7 to Friday January 11
Text: Milne's ``Algebraic number theory", Chapters 1 and 2.
Overview of the course. Basic notions of ring theory. The Krull dimension of a ring. Rings of integers.



2. Monday January 14 to Friday January 18
Text: Milne, Chapter 2.
On the Monday class we finished discussing integrality and gave the general definition of a Dedekind Ring.
Note: The Wednesday and Friday classes of this week are cancelled because I will be out of town.

Assignment 1 is now avalaible, and is due on January 30.




3. Monday January 21 to Friday January 25
Text: Milne, Chapter 3.
Dedekind rings. Discrete valuation rings. Unique factorisation into prime ideals.


4. Monday January 28 to Friday February 1
Text: Milne, Chapter 3.
Note: The Monday lecture of this week is cancelled because I will be out of town. In fact, because of a flight delay the Wednesday class was also cancelled! This means that I will be making up four extra hours in the coming months.

On the Friday lecture we discussed the theorem that the integral closure B of a Dedekind domain A in a finite extension L of its fraction field K is itself a Dedekind domain. We then discussed the factorisation of primes of A as a product of primes of B, proving that the degree of L over K is equal to the sum of the products of residual degrees and ramification degrees of the prime divisors of a prime of A.




Assignment 2 is now avalaible, and is due on February 20.



5. Monday February 4 to Friday February 8
Text: Milne, Chapter 3 and Kunz, Chapter 1
On Monday we discussed ramification further and showed that an extension B/A of Dedekind rings has finitely many ramified primes, by showing that these primes are exactly those that divide the discriminant of B/A. On Wednesday, we finished a few examples, and then launched on a brand new topic: an introduction to algebraic geometry following the textbook of Kunz.


6. Monday February 11 to Friday February 15
Text: Kunz, Chapter 1
We introduced the general notion of a variety and of the associated ideal and coordinate ring. The main theorems proved were the Hilbert basis theorem, asserting that a polynomial ring over a Noetherian ring is also Noetherian, and Hilbert's Nullstellensatz asserting that the ideal of polynomials vanishing on the set of rational points over an algebraically closed field of a variety determined by an ideal is equal to the radical of this ideal.



7. Monday February 18 to Friday February 22
TextKunz, chapter 1.
Note: This week we will have an extra lecture on Friday from 2:00 to 3:00 in room 1214.
This week was devoted to further generalities on linear and projective varieties, the notion of irreducible components and some basic results about intersections of projective varieties in projective space. For instance we showed that two projective hypersurfaces in n-dimensional projective space for n at least 2 always intersect.




Assignment 3 is now avalaible, and is due on Monday, March 11.



8. Monday February 25 to Friday March 1.
Text Kunz, chapter 1.
Note: This week we will have an extra lecture on Friday from 2:00 to 3:00 in room 1214.
This week was devoted to finishing covering Chapter 1 of Kunz, with a rudimentary introduction to the notion of scheme, building on what you learned last semester about the spectrum of a ring.





Monday March 4 to Friday March 8
Study Break





9. Monday March 11 to Friday March 15
Text: Knapp's Advanced Algebra, Chapter II.
The midterm will be held on Wednesday, March 13, at 8:30 AM, in class.
On Friday we will start on a new topic: the Wedderburn-Artin theory of semisimple algebras over a field.




Assignment 4 is now avalaible, and is due on Wednesday, March 27.



10. Monday March 18 to Friday March 22
Text: Knapp's Advanced Algebra, Chapter II.
This week we covered the Wedderburn classification of semi-simple algebras and of finite-dimensional simple algebras over a field.



11. Monday March 25 to Friday March 29
Text:Knapp's Advanced Algebra, Chapter II.

Note: This week we will have an extra lecture on Friday from 2:00 to 3:00 in room 1214.





12. Monday April 1 to Friday April 5
Text: Knappès Advanced Algebra, Chapter II.
Note: This week we will have an extra lecture on Friday from 2:00 to 3:00 in room 1214.


13. Monday April 8 to Friday April 12
Revision of the material covered in class.




The final exam will be held in class on Wednesday, April 24, from 9:00 AM to 12:00 PM, in BH920.