M 380 C Algebra 59000

CLASS HOURS: Mon, Wed, Fri 10:00 -- 11:00 RLM 9.166

OFFICE HOURS: Tue 9:00 -- 11:00 or by appointment.

TEXTBOOK: No textbook. See below for audio recordings of most of my lectures. Some useful references are: D. Dummit and R. Foote, Abstract Algebra, and S. Lang, Algebra. I used the following online notes:

• J. Milne's notes on group theory (local mirror).
• Abstract Algebra: The Basic Graduate Year by R. Ash. We will use Chapter 2 and Chapter 4 (local mirrors) (The Hilbert basis theorem is in Chapter 8).
• Notes on modules over PIDs (local mirror) by P. Garrett.
• P. May's write-up of R. Munshi's proof of the Nullstellensatz.
• My notes on Rings of fractions the hard way and Group-rings and polynomials

SYLLABUS: This the first half of the standard graduate Algebra course. We will cover Groups (groups, subgroups, quotient groups, group actions, direct products, solvable groups) and Rings (rings, domains, PIDs, UFDs, polynomial rings, modules).

PREREQUISITES: An introduction to abstract algebra at the undergraduate level.

EXAMS: A midterm on October 19th, 9:30AM in class and a final on Monday, 12/19 at 9AM, RLM 5.124.

HOMEWORK: I will assign homework on a weekly basis and it will be due following week, unless otherwise stipulated. Sometimes homework will be assigned out of old prelim exams which are available here.

GRADES: 20% homework + 30% midterm + 50% final.

Homework assignments:

1. Milne problems 42,43,49. Spring 00 prelim problem 2. Due 9/14.
2. Milne problems 37,51 and this problem. Due 9/21.
3. Milne problems 34,35,44,48,56,78 and this problem. Due 10/03 (note unusual date).
4. Spring 00 prelim problem 3, Fall 01 prelim problem 2, Spring 01 prelim problem 1, Fall 04 prelim problem 2. Due 10/12.
5. Do the problems on these notes on group-rings and polynomials (updated 10/26). Due 10/31.
6. Fall 99, prelim problem 1, Spring 01 prelim problem 2, Fall 02 prelim problem 2. Due 11/7. (I had also originally assigned Fall 03 prelim problem 2 but this was a mistake, I meant Spring 03, but never mind, just do the other three problems). Here are partial solutions to some of the problems and further comments answering questions from the review class.
7. Spring 03 prelim problem 2 and these two problems. Due 11/14.
8. Spring 00 prelim problem 4, Fall 03 prelim problem 4 (a,b,c) and these two problems. Due 11/21. Here is a solution to one of the problems.
9. Fall 01 prelim problem 3, Fall 05 prelim problems 2 and 3, and this problem. Due 11/30.

Audio recordings

• Aug, 31st
• Sept, 7th
• Sept, 9th
• Sept, 14th
• Nov, 11th
• Nov, 14th
• Nov, 18th
• Nov, 21st
• Nov, 23rd
• Nov, 30th
• Dec, 7th
• Free groups (date?)
• Solvable groups (date?)

Other links

• The wikipedia article on Category theory.
• An interesting paper on Category theory by B. Mazur.
• The Small Groups library.
• The Atlas of finite simple groups.
• The wikipedia article listing all finite simple groups.