# Mathematics 250A Fall, 2001 70 Evans Hall, TuTh 2:10-3:30

## Professor Ken Ribet

### 885 Evans Hall

Office telephone: 510 642 0648
Fax number: 510 642 8204
Secretary: 510 642 5026
email: ribet@math.berkeley.edu

## Textbook

Algebra by Serge Lang. You want the third edition, published by Addison Wesley Longman. This book is the classic algebra textbook for graduate courses. I used an earlier edition when I was an undergraduate at Brown University and a graduate student at Harvard. You can look at some unofficial companion material for Lang's book that was written by one of my colleagues. See, for instance, the errata to printings past and present.

## Syllabus

See the course description for general information about the syllabus. We will be studying several of the fundamental structures of abstract algebra, including groups, rings, modules and fields. I will try to cover Galois theory by the end of this semester. This course will continue with Math 250B in the spring semester, taught by Mark Haiman.

## Examinations

At each exam, you may bring in one standard-sized sheet of paper that summarizes theorems, formulas, definitions, examples, and other facts pertinent to the course. Please bring your own blue books or writing paper to the exams.

I taught this course once before, in 1992-1993. You are welcome to consult the archive for material from my old course, including the exams. Note that the course was taught on Mondays, Wednesdays and Fridays, so the midterms were only 50 minutes long.

The final course grade was a (monotone, non-decreasing) function of a single number between 0 and 200. In a message to students, I explained how this number would be computed:
```I'm working with the idea that the two midterms together are worth 50
points (in the sense that the maximum possible score is 50), that the
final exam will be worth 80 points and that the homework will be worth
70 points.  (I'll scale the scores given to me by Chu-Wee so that the
maxium possible homework score will be 70.)  In the grading scheme that
these three scores, say M + H + F, with the total of the scores being
between 0 and 200.

The new idea is to compute a second number: 2M + 10H/7, which will also
be between 0 and 200.  I'd base your final grade on the *maximum* of
these two numbers for you.
```
In the final exam, there were 7 problems, each worth 6 points. The maximum score was thus 42 instead of 80, so a scaling factor was used. The table that follows shows how the 40 registered UCB students did with respect to this scheme:

 SID mod 100 Total HW MT1 MT2 * Final Exam Grade Using Final Grade Without Final Max of two grades Grade 98 132.75 25 24 * 38 182.72 185.62 185.62 A+ 66 137.75 23 20 41 184.74 176.92 184.74 A+ 67 133.25 23 21 * 33 168.42 175.95 175.95 A+ 97 129.75 21 24 0 104.95 175.64 175.64 A+ 56 136 22 20 37 175.31 173.77 175.31 A+ 66 132.25 22 21 36 172.68 173.29 173.29 A+ 70 131 20 22 0 102.53 170.47 170.47 A+ 48 136.5 16 24 * 34 167.83 170.10 170.10 A+ 89 118.5 25 18 37 168.23 164.22 168.23 A 91 141.75 17 19 19 137.69 165.56 165.56 A 54 114.75 21 19 38 165.40 155.74 165.40 A 9 143.5 17 18 * 28 154.64 164.72 164.72 A 16 133 17 19 34 162.21 159.79 162.21 A 49 130 24 14 * 31 157.11 161.81 161.81 A 97 133.75 20 16 * 29 153.04 160.28 160.28 A 83 132.75 18 14 * 28 146.67 151.62 151.62 A 74 104.75 21 20 * 19 125.59 151.14 151.14 A 72 136.5 12 18 25 140.69 150.10 150.10 A 55 142 13 14 * 20 130.71 147.73 147.73 A 24 138.5 12 16 0 91.99 147.42 147.42 A 88 126.5 9 21 * 28 141.78 143.50 143.50 A- 70 77.5 21 12 39 143.09 117.16 143.09 A- 57 134.25 14 13 * 16 119.51 142.61 142.61 A- 12 96.25 17 22 0 83.47 141.53 141.53 A- 20 125.25 17 12 * 26 136.40 140.67 140.67 A- 52 108.5 18 14 0 82.13 135.62 135.62 B+ 73 97.5 16 13 31 133.10 122.36 133.10 B+ 71 113.25 16 12 * 26 129.85 130.75 130.75 B+ 81 135.75 11 9 * 19 118.91 129.60 129.60 B+ 85 69.5 25 16 * 15 101.68 127.87 127.87 B 91 104 16 13 * 13 101.81 126.65 126.65 B 50 99 14 16 12 98.60 125.35 125.35 B 43 118.25 12 11 0 77.64 124.05 124.05 S 74 93.5 17 13 * 21 113.20 121.72 121.72 B 12 67.5 21 11 29 118.43 108.55 118.43 B 6 65.25 15 22 * 24 112.86 117.07 117.07 B 77 67.75 14 10 0 55.30 92.72 92.72 P 10 60.5 11 13 18 86.24 87.93 87.93 B- 25 16.75 15 7 14 56.41 55.06 56.41 S 98 9 5 6 9 32.30 27.94 32.30 D

A * before the final exam grade means that the final exam grade listed is an upper bound. After I graded 6 out of the 7 problems, I stopped to check whether a full score on the remaining problem could make the "Grade computed using final" bigger than the grade without the final. If not, I decided that there was no point in grading the remaining problem but I awarded the student 6 points on the problem anyway.

For comparison, when I taught Math 250A in 1992, there were 26 students at the end of the semester. I gave out the following grades: 10 As, 13 Bs, 2 Cs and 1 S (satisfactory). In Fall, 2000, there were 18 students at the end of the semester in Math 250A. Half got As and half got Bs. More precisely, the distribution looked as follows:

```   A+  A  A-  B+  B   B-
2   4  3   2   3   4.
```

## Exclusive Anonymous Feedback Feature

During the semester, I maintained a "drop box" for student comments. ("Please let me know what I'm doing right and what I'm doing wrong. Constructive feedback is always welcome; don't hesitate to propose changes.") It should no longer be functional.
You can read the comments that were submitted during the course of the course.

## Homework

Homework will be assigned weekly. Problems will be graded by Chu-Wee Lim, the Graduate Student Instructor assigned to this course.
1. Assignment due September 4: Chapter I, problems 1, 3, 4, 5, 6, 7, 9. Possible solutions.
2. Assignment due September 11. Possible solutions.
3. Assignment due September 18. Possible solutions.
4. Assignment due September 25: Problems 32-41 on pp. 78-79. Possible solutions.
5. Assignment due Thursday, October 4: problems 44, 45, 49, 50, 52, 53 from Lang's Chapter I. Possible solutions and an alternative solution to problem 53 by Chu-Wee Lim are now available.
6. Assignment due Thursday, October 11: Chapter II, problems 1, 2, 3, 4, 5, 6.
7. Assignment due Thursday, October 18: Chapter II, problems 7, 10, 12, 13, 14, 15, 16. Possible solutions
8. Assignment due Thursday, October 25:
• Suppose that A is a commutative ring with identity. Let a be an element of A and let g(x) be a polynomial over A. Show that f(x) = a + xg(x) is a unit in A[x] if and only if a is a unit in A and some power of g(x) is 0.
• Problems from Chapter III: 6, 9, 10, 11, 12, 14, 15.
In problem 6, it seems clear to me that M is intended to be G-stable. Even with this assumption, however, the problem is apparently false. (This was explained to me by one of the students in the class.) Consider the case where G is the group of order 2 , S is the set {1,2}, and G acts on S non-trivially. The Z[G]-module Z[S] is Z x Z; G acts by flipping coordinates. Let M be the submodule of Z x Z consisting of pairs (a,b) with a and b either both odd or both even. It seems then that M has no Z-basis that is G-stable even though M is G-stable. Can you prove that this is the case?
If you want to see a cinematic proof of the snake lemma, watch "It's my turn." (This film is listed in Mathematical Fiction by Alex Kasman.)
Possible solutions to the homework problems and comments on our upcoming exam.
9. Assignment due Thursday, November 8:
• Read Noah Snyder's article, An alternate proof of Mason's theorem. (To download this paper, you need to be recognized as coming from berkeley.edu. If you're working from outside this domain, you can use the library proxy server if you have a UCB library card.)
• Chapter III: 17, 18
• Chapter IV: 5, 7ab, 7cd (these two parts are optional), 18
During the week of November 5, I starting writing up some solutions for this assignment, but never finished. You can at least look at the draft that I wrote during the week.
10. Assignment due Thursday, November 15: Chapter V, exercises 1, 2, 3, 5, 7, 8, 13, 18. Possible solutions that were prepared by Chu-Wee Lim.
11. Assignment due Thursday, November 29 (or Tuesday, December 4, at the very latest):
• Prove Corollary 1.4 on page 263 of the book (ok to use results that appear after that corollary if you don't make a circular argument)
• Chapter V, problems 19, 22 (do 21 first for yourself), 23a
• Chapter VI, problems 1bcdi, 4, 6
Possible solutions that were prepared by Chu-Wee Lim.
12. Assignment due Thursday, December 6 (or Tuesday, December 11, at the very latest): Chapter VI, problems 7, 8, 9, 10, 11, 15, 16

Kenneth A. Ribet , Math Department 3840, Berkeley CA 94720-3840