Fall, 2001

70 Evans Hall, TuTh 2:10-3:30

**Office telephone:** 510 642 0648

**Fax number:** 510 642 8204

**Secretary:** 510 642 5026

**email:**
`ribet@math.berkeley.edu`

- First Midterm: September 27, 2001; problems and possible solutions; distribution of scores.
- Second Midterm: November 1, 2001; problems and possible solutions. Distribution of scores:
- Final Exam, Thursday, December 13, 2001, 12:30-3:30 (exam group #5); questions and possible answers.

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.

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 I had in mind originally, your final grade would be based on the sum of 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 **A**s, 13 **B**s, 2 **C**s and 1 **S** (satisfactory).
In Fall, 2000, there were 18 students at the end of the semester in
Math 250A. Half got **A**s and half got **B**s.
More precisely, the distribution looked as follows:

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

- Assignment due September 4: Chapter I, problems 1, 3, 4, 5, 6, 7, 9. Possible solutions.
- Assignment due September 11. Possible solutions.
- Assignment due September 18. Possible solutions.
- Assignment due September 25: Problems 32-41 on pp. 78-79. Possible solutions.
- 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.
- Assignment due Thursday, October 11: Chapter II, problems 1, 2, 3, 4, 5, 6.
- Assignment due Thursday, October 18: Chapter II, problems 7, 10, 12, 13, 14, 15, 16. Possible solutions
- 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?

Possible solutions to the homework problems and comments on our upcoming exam. - 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

- Read
Noah
Snyder's
article,
An alternate proof of Mason's theorem.
(To download this paper, you need to be recognized as coming from
- 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.
- 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

- 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