189-235A: Algebra 1

Assignments

The bi-weekly assignments are an essential part of the course. You should plan to devote at least twenty hours or so (and quite possibly more) to the assignments. This means roughly ten hours a week. Not that delaying working on the assignment to the week right before it is due is not recommended, unless you happen to have a big chunk of free time in that week.

If you are stuck on a problem, you may seek out the help of a TA, the professor, or one of your classmates. It is OK to work on the assignments in groups, although you should hand them in individually. Do not neglect the assignments: experience shows there is a strong correlation between the work you put into them and how much you learn in the course, which will of course be reflected in your exam performance.

The assignments are normally due on Mondays at 2 PM at the latest. The assignments can either be handed in to me in class, or placed in the drop box on the 10th floor of Burnside Hall. and will be graded and returned to you on the Monday of the following week. Late assignments will not be accepted. The TA's will be working hard to grade your work and return it to you on the Monday of the following week, to ensure that you get prompt feedback before the handing in of the next assignment. With such a large class, they simply cannot afford to manage late asignments. Besides, it is not in your interest either to fall behind in your work.

Here are some guidelines for turning in your assignments.

All assignments:

1. should be presented as cleanly and clearly as possible. If your handwriting is less legible no you might want to consider typing your assignments.

2. Must be stapled individually (no paper clips, no tape, no origami, no plastic pockets, etc). Remember that there is a stapler in the 10th flor office where you hand back your assignments, should you need it.

3. Must be handed in on time. Remember, again, that late assignments will absolutely not be accepted.

4. Should present problems in order.

5. Should clearly bear the name and McGill ID of the student at the top of the first page.

Tips for writing proofs:

When writing up your proof, make sure that you explain your reasoning clearly and fully, using complete sentences. Mathematical notation, although admirable in its conciseness and power in many contexts, is no substitute for clearly written prose. Remember that in an exam or assignment, a wrong answer preceded by an almost correct, cogently argued justification will earn you (a lot of) extra credit. The same answer with no explanation of what led you there will earn you no points at all: the grader, unable to read your mind, will be forced to assume the worst...

It is always best to proceed linearly and in a logical order. For instance, say you want to prove the identity A=Z. If you know A=B, B=C, etc, write A=B=...=Z. This makes the proof easy to read as the reader only has to check each statement independently. In highschool some of you may have acquired the bad habit of writing the same proof by starting with A=Z, which you do not know to hold a priori (this already gives a hard time to the grader, who is left to guess whether you know what you are doing or are instead assuming what you were asked to prove), then proceed to write B=Y (since A=B, Y=Z this holds if and only if A=Z holds), C=W, ... up to L=M say. But you know L=M to be true so going up the chain this means A=Z. This is logically correct but very confusing and hard to follow as the proof is essentially written upside down. I did not penalize anyone for this on the first assignment because MATH 235 is one of the first proof based courses but make sure to write proofs where, proceeding from the top to the bottom, every statement can be deduced from the previous statements. Finally, some of you seem to be fervent advocates of the LHS, RHS school of proof writing. Note that if one can proceed linearly to show LHS=A=B=...=Z=RHS one should try to avoid showing that LHS=A=...=Z=Z', RHS=A'=B'=...=Z' which implies LHS=RHS. If lenghty computations are involved and one has difficulty using the first method of proof then one is justified in using the second.

Assignment 1 (pdf). Due: Monday, Sept. 23.

Five questions in Assignment 1 were graded, out of 20 points each, as follows:
Question 2, 10 points for (b) and 10 points for (d).
Question 4, 10 points for getting the correct formula, and 10 points for a decent proof. 5 points for a proof that has some issues but is still going in the right direction.
Question 5, 10 points for (a) and 10 points for (b)
Question 6, 6 points for (c), 7 points for (d), and 7 points for (e)
Question 7. Only (b) was graded, out of 20.

Note that in particular, since not all questions were graded, it is important that you have a look at the solutions below to make sure that your work in the other problems was correct.

Solutions to Assignment 1
Assignment 2 (pdf). Due: Monday, Oct. 7.

Here is the marking scheme for Assignment 2.
The following 5 questions were graded, each out of 20 points, as follows:
Question 1. 5 points for showing R is a ring. 5 points for showing 3 is irreducible. 10 points for the counterexample to the Gauss Lemma.
Question 2. 5 for each of a,b,c,d.
Question 5. 10 points for the "prove or disprove..." part. 10 points for the example.
Question 8, 20 points.
Question 9, 20 points.
Solutions to Assignment 2
Assignment 3 (pdf). Due: Monday, Oct 21

Solutions to Assignment 3
Assignment 4 (pdf). Due: Monday, Nov. 4
Solutions to Assignment 4
Assignment 5 (pdf). Due: Wednesday, Nov. 20
Solutions to Assignment 5

There will be no assignment 6, for lack of time. Instead, I propose that you work through the following Practice Finals, which will not be graded. You should work through them as part of your studying for the final exam, and you can (and should) bring up any questions that remain at the review sessions.

(Click here for practice finals).

To be discussed in the last week of lectures.