Mathematics 250A
Fall, 2004

Lectures

Syllabus

Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.

Textbook

Graduate Student Instructor

Grading

The table below shows the distribution of scores for the 25 registered UC students in the course. It includes a line for the fictional student "Gauss," who had perfect scores. There were also three students in the course who do not have student IDs.

SID mod 100	MT1	MT2	HW total	Final Exam	Composite Score	Letter grade
0	30	30	420	40	100.00	Gauss
3	19	23	383.67	32	80.41	A
11	14	7	294	22	53.50	B
14	30	25	353	36	88.71	A+
17	12	18	297.67	22	58.26	B+
26	17	10	277	20	53.29	B
29	5	17	259	13	42.50	C
40	24	17	312	24	66.79	A-
43	14	21	366	24	67.64	A-
45	11	5	195	7	28.93	D
48	22	26	411	34	87.36	A+
58	12	11	353	13	49.71	B-
63	30	23	342	30	80.93	A
63	18	21	408	23	71.64	A
65	19	27	388	24	74.71	A
68	14	19	400	14	59.07	B+
68	26	17	341	28	73.86	A
71	12	7	133	20	39.00	C
77	21	15	404	39	85.86	A+
79	17	21	352	35	79.14	A
82	22	25	349.33	36	84.45	A+
85	19	26	362	36	84.36	A+
90	19	13	350	17	58.00	B+
92	18	11	238	19	50.50	B-
95	9	6	263.33	10	36.31	C-
98	12	8	284.67	18	48.33	B-

Examinations

• First midterm exam, September 30 [questions and possible answers; scores]
• Last midterm exam, November 4 [questions and possible answers; scores]
• Final examination, December 20, 2004, 12:30-3:30PM in C125 Cheit [questions and possible answers; scores]

I taught this course three years ago. You can look at the web page for my old course for more information (including exam questions and solutions).

Homework

1. Assignment due September 7: Chapter I, 4 (ignore the hypothesis that K normalizes H), 5, 6, 7, 8, 9. In Problem 8, there is a pair of misprints: as the problem is written, there are three union signs, where the indices are respectively c, x_c and x_c. The first union should be over elements x_c; these form a set of representatives for the coset space H/H", where H" is the intersection of H and the conjugate of H' by c. The second union is over elements c. The third union is as written; namely, it's a union over the same set of elements x_c that appeared in the first union. In short, one needs to exchange the indices in the first and second union signs. (Possible solutions, written by Ribet.)
2. Assignment due September 14: Chapter I, problems 13, 14, 16, 15, 17, 19, 20, 22, 23bc. I wanted to assign Problem 12, but it's sort of a mess. I wrote up some comments on that problem and took photos of the two pages (p. 76 and p. 77 where the problems appear. (Possible solutions, written by Ribet.)
3. Assignment due September 21; this was last updated on September 19, 2004. (Possible solutions, written by Ribet and updated on September 22, 2004.)
4. Assignment due October 5: Chapter I, problems 24, 25, 26a, 28, 29, 30 39, 40, 41, 46, 47, 48, 50, 52 (Possible solutions, written by Ribet).
5. Assignment due October 12: Chapter II, problems 1-7 (possible solutions, written by Chu-Wee Lim).
6. Assignment due October 19: Chapter II, problems on Dedekind rings (13-19). There are some relevant comments on the comments page and now possible solutions, written by Ribet.
7. Assignment due October 26. For the last problem, you should probably read Chapter VII through to the statement of Proposition 1.1, which occurs at the top of the third page of the chapter. Also, note that an "integral ideal" of a ring is the same thing as an "ideal" of the ring; people sometimes use the adjective "integral" to stress that they're not talking about fractional ideals. (Possible solutions, written by Ribet.)
8. The assignment due November 9 is like HW #4 in that it represents three lectures, rather than two. Note that there are possible solutions, written by Chu-Wee. For a slightly different discussion of problem 13c in Chapter III, you could consult page 11 (and page 12) of "Introduction to Algebraic K-Theory" by John Milnor. In the book, the main theorem that emerges in problem 13c is attributed to Steinitz.
9. Assignment due November 16: Chapter IV, problems 3, 5, 6, 7, 18. In problem 7, p is the characteristic of k, so that q is a power of p. (Possible solutions, written by Ribet.)
10. Assignment due November 30: Chapter V, problems 3, 5, 7, 9, 11 (all parts). (Possible solutions, written by Ribet.)
11. Last assignment, due December 9: Prove Corollary 1.4 on page 263. (It's not "obvious," contrary to what our author says. You can appeal to results that come after the corollary if you have checked that the statement of the corollary is not used in the proofs of the subsequent results.) Also, do the following problems from Chapter VI: 1 (a through e), 5, 6, 7, 9, 11, 15. (Possible solutions, written by Ribet.)

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Added Christmas Day, 2004: comments are all finished now. Thanks for your helpful feedback and questions.