The final examination is administered by the Faculty of Arts and Sciences. Follow the link for the schedule, regulations, and any logistical questions.
The final exam will be a "traditional'' three-hour final exam. Unlike Tests 1 and 2, it will not have a special format.
All questions on the final exam will be directly about Part 4 of the course (field extensions and Galois theory). This is because in order to do well on this part, you need to understand the previous parts. So, for example, you need to understand group theory to compute Galois groups of field extensions; and you need to understand about ring theory, ideals, factorization, and polynomials for many of the proofs involved in field extensions.
I am available for consultation between the end of classes and the exam period. Just email me for an appointment and I will be happy to meet with you.
All announcements and course materials will be posted here on this website.
Grades will be posted on blackboard.
Do you have any comments or complaints that you would like to submit to me anonymously? If so, you can do it here .
(If you do not mind doing it nonymously, talking in person or sending an email are good options.)
HOMEWORK ASSIGNMENTS AND CALENDAR
I encourage you to attempt the reading assignments before the lectures on that topic start.
I will post every homework assignment here at least one week before it is due. I will not update them without warning less than a week before they are due.
I expect you to do all the problems in the homework set, but only the ones in bold and brackets are to be turned in on the day the homework set is due. Sometimes the not-to-be-handed-in problems will help you solve the to-be-handed-in problems. They are due at the beginning of the class. I will not accept late assignments.
PART 1: Basic theory of groups.
Mon Sep 8 -- Mon Sep 15
Reading:
Section 0.1. - These are basic concepts that you should know before the course starts, and that I will not cover in lecture.
Section 0.2. - I will come back to cover this later in the course, but it will still be beneficial to read it now.
Some extra links for those interested in defining groups via presentations:
If you want to see an example of a group presentation whose word problem is undecidable, see this paper. The example given has 10 generators and 27 relations.
If you are curious to learn more about how (computationally) hard it is to understand abstract groups given by generators and relations, you can read the first section of this paper.
This may not make sense unless you have an interest in computer science or in logic and foundations.
If you find Burnside's Lemma interesting, you may want to learn about its generalization, the Redfield-Polya Theorem. BL answers the question "In how many different ways can we colour the faces of a cube with red, white, and blue?" RPT answers the follow-up question "And how many of those have exactly R red faces, B blue faces and W white faces, for each triple (R,B,W)?" Here is a paper to learn more.
Also review sections 1.7, 2.2, 4.1. These sections contains the basis of group actions. We have already seen all these topics in bits here and there, and I won't repeat them in class. I encourage you to review them in preparation for the following week.
Homework #5 (due on Friday, October 17):
Section 2.2: problems 6, 7, 10, [12] (this action is necessary in the definition of the alternating group).
Visualizing the groups of symmetries of platonic solids.
This website may help visualize the group of symmetries of a dodecahedron or an icosahedron, and understand why they are isomorphic to A_5. Notice that you can rotate the images with your mouse. Here is another link.
If you have trouble visualizing the platonic solids, you can get a cheap set of dice that includes all five of them from any comic book store.
Microbes have interesting groups of symmetries (ignore the happy faces; the real things are mere blobs with protuberances). What are the groups of symmetries of the common cold virus, HIV, herpes, hepatitis, and mono?
Here is a table with the number of isomorphism types of groups with order up to 2000. (Link)
A big example. As an example of a very involved classification, I have written the classification of groups of order 60 up to isomorphism. You may want to postpone reading this until you are done reading the textbook and working on the homework. (Link)
Canonical forms. In some classifications, it is useful to be able to construct explicitly a matrix of a certain order of a certain size with coefficients on a certain field. (For instance, how do you construct a 3-by-3 matrix with coefficients in Z/2Z, and which has order 7?) One approach to do this is to use the Canonical Forms for matrices. This would normally be learned in MAT247. Just in case, I have written some brief, sketchy notes about it. Strictly speaking, you do not need this to solve your project problem, but it is a useful tool that sometimes provides shortcuts. (Link)
The culmination of this part of the course will be the project. See information here.
PART 3: Ring theory.
Mon Nov 24 - Fri Dec 3
Read sections 7.1, 7.2, 7.3, and Appendix I
Homework #9 (due on Friday, January 9): Note: At the beginning of the problems of each section of Chapter 7, there are some notation conditions that apply to all problems in that section.
Some students asked about the proof of the Banach-Tarski ParadoxTheorem. This is only vaguely related to our course (the proof uses both the Axiom of Choice, free groups, and group actions) but for those interested, here are some notes I wrote for a talk I gave in the past about it.
Mon Jan 5 - Mon Jan 12
Read sections 7.4, 7.5, 7.6
Homework #10 (due on Friday, January 16): Note: At the beginning of the problems of each section of Chapter 7, there are some notation conditions that apply to all problems in that section.
Once you understand the proof that every non-zero ring with identity has a maximal ideal, see if you can follow the rap version of the proof of existence of maximal ideals. Care to give it a try? (Proof due to Ari Nieh.)
To build one example of a ring without maximal ideals, see these brief exercises. If you want to read further on the topic, see these paper.
[Mon Jan 12 - Mon Jan 19] AND [Fri Jan 23]: Unique-factorization domains.
Reading: For the first four days, you have two options:
Read sections 8.1, 8.2, 8.3 from the book. Notice that the order of topics and emphasis is different, but of course, the explanations in the book are still entirely fine.
On Friday, January 23 we will study factorization in the Gaussian integers. Either read pages 289-292 on the book, or follow along with the worksheet below.
Reminder: your referee report on the project is due on Friday, January 23.
A propos of it, cheer up with non-abelian. Watch closely, as you may recognize someone. (In case you are unfamiliar with the original by Simon and Garfunkel, here it is.
The material for this last part of the course corresponds to Chapters 13 and 14. The book goes into much more depth than we will, so I will provide my own notes instead. I will keep the latest version of the notes here and I will annotate any time there is an update.
Splitting fields, normal extensions, and algebraic closures (section 5 from my notes OR section 13.4 from the book).
Separable extensions (section 6 from my notes OR section 13.5 from the book; if you follow the book, only till page 549 plus Corollary 39 -- we will postpone finite fields till later).
Statement, proof, and examples of the Fundamental Theorem of Galois Theory (section 7 of my notes OR section 14.1-14.2 in the book; the book covers this material in a very different way and goes much more in depth. However, the examples in the book are very useful: pages 559, 560, 563-566, 576-581.)
Mon Mar 30 - Wed Apr 1: Extra topics (not included in the final exam).
An introduction to Differential Galois Theory.
There are two motivating questions: 1) which elementary functions have elementary antiderivatives?, and 2) which differential equations can be solved using elementary functions and antiderivatives only? (we did not talk about the second question).
Here is a one-page summary of the main concepts and results.
Here are some notes by Brian Conrad which fully prove the impossibility of writing some antiderivatives in terms of elementary functions. It is 13-page long and I believe you have all the prerequisites necessary to understand this exposition.
On the history of solution of polynomial equations
Most of the story I told you in class can be found in the excellent history of mathematics archive at St Andrews University in Scotland.
You may be particularly interested in the biography of Tartaglia and in the article "Tartaglia vs Cardano", which includes plenty of quotes from their actual correspondence, as well as links to biographies of the other characters in this play.
