Comment: If you search for "abelian grape" on google, you'll get lots of hits with links to more math jokes.
Ribet's answer: Because I'm teaching two courses this semester, I simply don't have the time to post solutions to all the homework problems. What I can do is write up solutions to a few problems each week -- the ones that are most requested. If there's a problem that you want to see written up, let me know (either by e-mail or through this comments page). Someone asked me how to do problem #23 in the homework that was due this week, so I put up a solution.
Ribet's answer: As far as I'm concerned, the "course" comprises everything that transpires between us, including homework and lectures. I actually do have some history of making examples from my lectures come back in exam questions. On the other hand, if you want to know whether a question or topic would be reasonable for a midterm, put yourself in the position of the examiner (me). If a question covers material that is only incidental to the main topics, it's not a likely question. If a question will stump most people in the class, it's not a good one. If a question will elicit answers that are hard to evaluate, then it's probably a bad question. Use your own judgement.
If some example seems to go on and on during a lecture, you should stop me to get me to explain why I'm droning on about the example and to connect up the example to whatever it was that we were doing before the example started.
Reply: I've asked the grader to get them back to me sooner rather than later. I have one assignment (#2, I guess) already; it will be returned in class on Tuesday. Students who come to my office today (Saturday, February 15) around 4PM will likely find me there and can get their papers ahead of time.
Reply: I wouldn't think so. That question pertains to the Chinese Remainder Theorem, which can be discussed (for integers) soon after a discussion of the Euclidean algorithm. In our textbook, there's no special discussion of the CRT for integers. The CRT is proved for arbitrary rings starting around page 266.
Reply: There wasn't enough sentiment for it. It was clear to me in class that the idea was dead. The conclusion was that I would include a "cheat sheet" on the exam itself and that students would send me e-mail with wish-lists of points to include on the cheat sheet. I got only one message, which asked for a list of definitions of concepts like "center" and "normalizer". I decided not to have a list like that but instead to recall definitions as needed in statements of problems.
If they're tedious, you're not doing them right. :-)
OK, we'll try for fewer.
Reply: I have actually done this on and off. This is the first feedback that I've received -- good that it's positive! I haven't sent out e-mail messages consistently because I didn't want to flood people's mailboxes. Maybe your mailboxes are flooded already; mine are, by spam.
On Tuesday, March 4, we will discuss Theorem 20 on page 100, touch briefly on the material in § 3.4, and then concentrate on the definition of the sign of a permutation, which is introduced in § 3.5.
Reply: Yes, a line of html code was repeated by mistake. This has been fixed now; thanks for pointing it out.
(No, there won't be homework due next week. The course web page shows the due dates for the remaining homeworks of the semester. After this week's, there will be four assignments left to go.)
I really don't have time to write-up full solutions to the homework assignments. What I can do is to write up solutions to a few exercises that people are especially eager to see. The "comments" that are associated with recent assignments are there because students asked me about specific problems. It's quite possible that I will have more homework back from the grader at my office tomorrow, by the way.
Reply: Although the vast majority of grades are now filed electronically, there is currently no mechanism for instructors to get statistical information back out of the Bear Facts system. The only way to find out historical grade distributions is to pore over old grade sheets. With the help of the staff in 970 Evans, I was able to look at the grades for H113 over the last three years (spring 2000, 2001, 2002). There were a total of 31 letter grades given out over those three years. (Enrollments were a tad higher, so some students must have gotten incompletes or P/NP grades.) The distribution was roughly 39% A, 35% B, 16% C, 0% D, 10% F. In my experience, a lot of the students who get Fs in class are students who left the class but did not drop officially for some reason. It's hard to make much of these statistics because the numbers involved are so low. Our class has 28 students, by the way, one of whom is taking the course by concurrent enrollment through UC Extension. (Grades of students who register through concurrent enrollment don't appear on the grade sheets.)
Reply: I hadn't formulated a policy. I suspect that most students would prefer a policy with N=1. Last semester in Math 110, I took N = 1.5: I disregarded each student's lowest HW score and counted only half of each student's next-to-lowest score. If there were 14 homeworks, each counting 10 points, the highest possible homework score was then 125. In this course, I'm amenable to a policy with positive N if the maximum possible score each week has been constant. If the grader awarded points out of a possible 37 one week and 43 the following week, this will complicate matters. If you have strong feelings on this matter, please let me know what you think.
Reply: There are binders in 970 Evans that you can consult. These contain summaries of a professor's evaluations in courses; the summaries are sorted alphabetically by professor. It takes a while for summaries to show up after a course is over -- a staff member reads the students' comments and attempts to summarize them in a couple of paragraphs.
Reply: OK, I agree: it's easy enough to normalize the HW scores by multiplying them by a coefficient. After this is done, there's still the question of N (the number of low scores that get dropped). Averaging all scores takes N=0. This is a perfectly reasonable value, which allows students who do well on all 14 homeworks to score better than students who score well on 13 out of 14. Deciding that students won't be penalized for a bad day means that students can't be rewarded for having no bad days.
Reply: The gcd is 1403109613 and the coefficients a and b are 277522 and -32573. I got this from gp (see PARI/gp Central), using their bezout function. Anyway, I said only that we'd try for fewer tedious problems. I didn't say that we'd succeed in eliminating them entirely.
Reply: I did this yesterday. Hope the comments are comprehensible and helpful.
Reply: In exercise 3 in 9.2, the assertion seems logically correct to me even if f(x) is a constant polynomial, so there is no reason to exclude this case. If I understand correctly, the definition makes it true that constant polynomials are never irreducible, so you'd have to show that F[x]/(f(x)) is never a field when f(x) is constant. For the second question, F[x]/(x+1) is isomorphic to F under the map that takes a polynomial g(x) to its value at -1. The polynomial x^2+1 is mapped to 2 under this map, so its inverse is 1/2. You can see what's going on a bit more explicitly by saying that x^2+1 = (x+1)(x-1)+2, so x^2+1 is the same thing as 2 in the quotient.
Reply: I'll be able to do this, but not for a few days. Expect to see solutions early next week.
Reply: My view is that it's the student's responsiblity to figure out what sections of the book we have studied in the class. The final covers the course, and your job at the end of the semester is to review the course as a whole.
Reply: Finals do tend to place some emphasis on material from the last third of the semester, but they're supposed to do a pretty good job of covering the whole course.
Reply: Whatever: the problem just asks for an example, and the solution supplies one. If you prefer 5 to 7 because 5 is a smaller number, that's OK with me.
Reply: My life is busy enough that I haven't even started grading the exam. So don't expect to find out your grades instantly. When I've computed the grades, I will post them electronically to Bear Facts, so you'll be able to get them on line. I'll also insert into our class home page some kind of spreadsheet that shows how grades were computed. If you look at the Math 250A home page from three semesters ago, you'll see an example of the sort of table that I'm talking about. I will list students non-alphabetically along with their grades and the last two digits of their SIDs. This will enable you to locate yourself pretty quickly but will not make it possible for random people to see how you did.
As far as the HW goes, I will weight all weekly homework assignments equally (by dividing grades by their maximum possible values) and then subtract off the lowest single score.