University of Wisconsin-Madison
University of Wisconsin-Madison
Mathematics Department

Math 741
Abstract Algebra
Lecturer: Arun Ram

Fall 2001

All parts of the following may be negotiated by the client. In the absence of negotiation it will become immutable law (as always, subject to change without prior notification).

Math Dictionary for UW Math Graduate Students !! Try it!!

Qualifying exams on-line

About the course: Math 741 is the first graduate course in abstract algebra. The core topics are groups, rings, and fields. The main goals of the course are

(1) to give a good grounding in basic algebra needed for future research and applications of algebra and

(2) to prepare you for taking the qualifying exam in algebra.

Though it is not necessary, some previous experience with abstract algebra, such as our undergraduate abstract algebra course (Math 541), would be helpful for doing well in this class. A good study ethic is a must.

Lectures: There will be two 75 minute lectures per week:

Tuesday 8:00-9:15am and Thursday 8:00-9:15am in B105 Van Vleck.

I apologize for the fact that this time may not be the best hour, but it is crucial for our department to schedule this core course at a time when it doesn't conflict with other core courses. Also this time of the morning is probably the best time of the day for me (provided I've had a good strong coffee before hand) and so you're probably getting a good product from me at this hour.

Text: I will not lecture or choose problems from any single book. In preparing my lectures I, myself, will make heavy use of the following books:

Algebra, A graduate course, by M. Isaacs, Brooks/Cole Publishing Co., 1994, ISBN: 0-534-19002-2.

Algebra, by M. Artin, Prentice Hall Inc., 1991, ISBN: 0-13-004763-5.

Algebra Chaps 1-7, by N. Bourbaki, Translated from the French,
Springer-Verlag, Berlin, 1990, ISBN: 3-540-19375-8 and ISBN: 3-540-19373-1.

Algebra, by S. Lang, Second edition. Addison-Wesley Publishing Company, 1984, ISBN: 0-201-05487-6.

Algebra, by T.W. Hungerford, Graduate Texts in Mathematics, 73,
Springer-Verlag, New York-Berlin, 1980, ISBN: 0-387-90518-9.

Introduction to commutative algebra, by Atiyah and Macdonald, Addison-Wesley, 1969.

Polynomial invariants of finite groups, by D. Benson, London Mathematical Society Lecture Note Series, 190,
Cambridge University Press, Cambridge, 1993, ISBN: 0-521-45886-2.

Any combination of these books would be extremely helpful study tools. I strongly recommend the book by Isaacs (read it, study it, do problems from it, look things up in it, absorb it, osmose it, sleep with it under your pillow, and carry it with you everywhere you go).

You may also find the information on the Math 541 web page to be a useful tool for helping to learn the material for this course.

Office hours: by appointment. Come up before or after class and make sure that I write your name and a time to meet in my date book. If you are having any problems, questions or concerns please come and to to talk about it. If you do not keep me informed I cannot help.

Extra meeting: I believe it's extremely important for the first year graduate students to discuss and get advice on the qual problems and other things (jobs, futures in mathematics, how to do research, how to choose an advisor, ...) in a more relaxed setting. The best way that I can think of to do this is to schedule one extra meeting per week (probably 5-7pm on Tuesday afternoons) for these discussions.

Requirements for the course:

1. Colloquium: One of the most important things that first year graduate students need to do is to get a picture of the world of math as it is now (not as it was 50-100 years ago, which is what we teach in Math 741, 721 and 751). The best way to do this is to go to the colloquium every week. I myself go to the colloquium every week and I will be pleased to see the Math 741 students going as well.

2. Mentoring of Math 541 students: I will also be teaching the undergraduate level abstract algebra (Math 541) concurrently. I will try to set up a bijective correspondence between the students in the 741 class with the 541 students so that they will always have someone to go to for additional help/advice etc. This should be beneficial to the students in both classes, because (1) it will help the graduate students in Math 741 review and strengthen their Math 541 skills by having to explain it, and (2) it will give the Math 541 students another good (and more personal) resource for learning and clarifying the material.

3. Homework and exams: There will be no homework for the class per se--you are responsible for making sure that you are able to do well on the exams for the course. Your grade will be based on the exams. The exams will be as follows:

After month 1: One prespecified past qualifying exam in algebra.

After month 2: A selection of problems from two prespecified past qualifying exams in algebra.

After month 3: A selection of problems from four prespecified qualifying exams in algebra.

You should study and do problems to prepare yourself for taking the qualifying exam. I am always happy to grade solutions to problems that you do in order to help you receive feedback on your solutions.

Syllabus: The following is some general plan for the topics to be covered in the Math 741-742 sequence. I have not yet decided on the ordering of topics.

(1) Groups, rings, modules, vector spaces, algebras, Lie algebras. Homomorphisms, isomorphisms, kernels, images. Categories, abelian categories.

(2) Objects satisfying a universal property: quotient groups, direct sums, direct products, free groups, free modules, tensor products, tensor algebras, symmetric algebras, exterior algebras, inverse limits, direct limits, polynomial rings, group algebras.

(3) Generators and relations.

(4) Examples of groups: Cyclic groups, dihedral groups, symmetric groups, alternating groups, reflection groups, matrix groups, braid groups.

(5) Examples of rings and algebras: matrix rings, polynomial rings, Laurent polynomials, formal power series, integers, p-adic numbers, quaternions, quadratic algebras, Cayley algebras, Clifford algebras, tensor algebras, symmetric algebras, exterior algebras, group algebras, Brauer algebras, Gaussian integers.

(6) Orbits, Stabilizers, Centralizers, Normalizers, Conjugacy classes, centers, Sylow Theorems.

(7) Composition series, lower central series, derived series, Schreier's refinement theorem, the Jordan-Holder theorem, nilpotent groups, solvable groups, supersolvable groups, p-groups

(8) Finitely generated modules over a PID, finitely generated abelian groups, canonical forms, Jordan normal form, conjugacy classes in matrix groups.

(9) Polynomial rings, derivatives, derivations, completions, formal power series.

(10) Quotient fields, finite fields, p-adic fields, number fields, reals, complexes, quaternions. Extensions.

(11) Galois theory.

(12) Factorization: PID, UFDs, the Euclidean algorithm, Dedekind domains.

Other notes: The Math Department Faculty Minority Liaison is Prof. Daniel Rider. He has information available concerning diversity and multicultural issues (e.g. support services, academic internships and grants/fellowships). He is also available to discuss minority students' concerns about mathematics courses. Prof. Rider can be reached at 263-3603, or in 821 Van Vleck. See the web page at http;//