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

Math 541
Modern Algebra
A first course in 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).

About the course:

Math 541 is the first course in abstract algebra. The core topics are groups, rings, and fields. Math 541 is particularly useful for future K-12 math teachers since one of the main points of this course is to explain where addition, subtraction, multiplication and division come from, why they do what they do, and how they can be sensibly modified. If you are going to be teaching math, then you will need to explain these things to your students. Along with Math 521 and Math 551 this course is a necessity for students considering going on to graduate school in mathematics. In order to do well in this course it will help to have (1) a good study ethic and (2) some experience with matrix algebra, such as that obtained in Math 340 or Math 320 (or any one of several other math, engineering or economics or statistics courses).

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

Tuesday 9:30-10:45am and Thursday 9:30-10:45am in B105 Van Vleck (Note that this is a change from the originally scheduled room).

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 in to talk about it. If you do not keep me informed I cannot help.

Extra meeting: One of the goals of this course is to teach everybody how to construct and write proofs. The best way to learn this is to actually see how a proof evolves i.e., the process of producing a proof. We will schedule one extra meeting per week (probably 5-7pm on Thursday afternoons) for discussing, learning, and absorbing the techniques of constructing good proofs. This way I can, in a slow relaxed setting, show you how its done.

Graduate student mentors: I will also be teaching the graduate level version of the same course (Math 741) concurrently. I will try to set up a bijective correspndence between the students in the 741 class with the 541 students so that you 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 personal) resource for learning and clarifying the material.

Text: There is no textbook for the course per se, as I do not lecture or choose problems from any single book. In preparing my lectures and choosing homework problems I, myself, will make heavy use of the following books:

Course notes that I wrote myself several years ago.
I will have these copied and make them available (for whatever it costs for me to xerox them).

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

Cours d'Algebre (French)[Course in Algebra], by R. Godement, Third edition, Hermann, Paris, 1987, ISBN: 2-7056-5241-8.

Topics in Abstract Algebra, by I.N. Herstein, J. Wiley and Sons, 1975, ISBN: 0 471 01090 1.

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.

Any combination of these books would be extremely helpful study tools. I strongly recommend the book by Artin (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).

Syllabus: The following is my general plan for the topics to be covered and the ordering which I have in my mind at the moment.

(1) Definitions and examples of groups, rings, fields, modules, vector spaces.

We will make lists of the standard examples and discuss some of them in detail.

(2) Generators and relations and finite groups of low order. More examples: Cyclic groups,
Dihedral groups, symmetry groups of polytopes, tetrahedral groups,
octahedral groups, icosahedral groups, the Buckyball.

(3) Subgroups and cosets.

(4) Families of groups: Cyclic groups, Dihedral groups, Symmetric groups,
Alternating groups, and matrix groups.

(5) Orbits, Stabilizers, Centralizers, Normalizers, Conjugacy classes and centers.

(6) Homomorphisms, kernels, images, normal subgroups.

(7) Quotients (and more normal subgroups, kernels and images) and homomorphism theorems.

(8) Sylow Theorems

(9) More examples and groups of low order.

(10) Review of definitions of rings, fields, modules and vector spaces.

(11) Examples of rings: Matrix rings, polynomial rings, group algebras, Brauer algebras.

(12) Homomorphisms, kernels, images, ideals.

(13) Examples of ideals and quotients -- integers, polynomial rings, upper triangular matrices.

(14) Examples of fields: R, Q, C, finite fields, p-adic fields, number fields, quaternions.

(15) Prime ideals, maximal ideals, integral domains

(16) PIDs and the Euclidean algorithm. Examples: integers and polynomial rings.

(17) Review of row reduction for matrices. Normal forms.

(18) Relationship of normal forms to modules for PIDs

(19) Rational canonical form, Smith normal form and Jordan canonical form.

(20) Finite generated modules over PIDs and the fundamental theorem of abelian groups.

This accounts for 28 lectures (over 14 weeks) with some flexiblility and time for review sessions etc.

Grading: The term grade will be based on homework and the exams as follows: Homework: 25% Vocabulary quizzes: 25% Midterm: 25% Final Exam: 25%. Final grades are computed by totalling the points from the homework, the midterms and the final. Grade letters will be assigned with the following curve as a guideline: 20% A's, 30% B's, 30% C's, 20% D's and F's.

Homework: Homework will be due weekly. You will be required to find your own problems (from any of source of your choice) and provide careful solutions to the problems you submit. We may also set up a system by which you "grade" (i.e. mark and make suggestions on) homeworks of your fellow students--with the final grading process on each homework and the assigning of points done by me or the grader.

Vocabulary quizzes: One of the challenges when one is first starting to learn proof oriented mathematics, is to remember the rules of the game (the definitions). This EXTREMELY IMPORTANT part of doing proofs is easy since it is just regurgitation. To keep everybody in shape on this we will have a vocabulary quiz during the first 5 or 10 minutes of each class period (a 10 minute quiz on Tuesdays, and a 5 minute quiz on Thursdays).

Exams: There will be a midterm exam and a final. I (with the help of the students from Math 741) will compile a list of the homeowrk problems submitted. The exam problems will be chosen, verbatim, from these lists. The midterm exam will be in class on October 25. There will be a 2 hour final exam at 2:45pm on Wednesday December 19. The exams will be "regurgitation" from the notes handed out in class and the homework problems in Artin. The final will be cumulative.

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;//

Problem lists:

Week 1: Submitted before September ???, 2001 (The following links are not yet active) postscript file or pdf file

Week 2: Submitted before September ???, 2001 (The following links are not yet active) postscript file or pdf file

Downloading/printing. I have successfully dowloaded the homework at the computers in the Computer Lab in Van Vleck 101. This is a computer lab for students so please take advantage. You will need a red debit card to print in the Van Vleck 101 computer lab. Ask the consultants there in the lab where to find a machine to buy a debit card. The following information may also be helpful.

(1) .ps files can be sent directly to a postscript printer or viewed on the screen with a previewer such as Ghostview. Postscript printers are available in most campus computer labs, for example the Computer Lab in Van Vleck 101.

(2) .pdf files can usually be read with Acrobat reader. You can down load Acrobat Reader free from the following location: Free software from Adobe.

(3) In the Van Vleck 101 Computer Lab you can save the postscript file to the desktop and then open it. This will create an Acrobat file which you can then open and print. The consultant in the lab can also show you how to do this. They are very nice and helpful.