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

Math 541
Modern Algebra
A first course in abstract Algebra
Lecturer: Arun Ram

Fall 2003

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).

Special goal: One of the goals of this course is to teach everybody how to construct and write proofs.

Lectures: There will be three 50 minute lectures per week:

MWF 9:55-10:45 in B113 Van Vleck

Text: The textbook for the course is

Abstract Algebra, by David Dummit and Richard Foote, Second Edition, John Wiley and Sons, Inc., 1999.

I also recommend the following books:

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

Basic Abstract Algebra, by P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Second edition, Cambridge University Press, 1986.

I strongly recommend you get one or all of these books (read them, study them, do problems from them, look things up in them, absorb them, osmose them, sleep with them under your pillow, and carry them with you everywhere you go).

Course Notes:

How to do proofs: postscript file or pdf file

Sets and functions: postscript file or pdf file

Groups and Group actions: postscript file or pdf file

Groups and Group actions, the proofs: postscript file or pdf file

A list of small groups: postscript file or pdf file

Cyclic groups: postscript file or pdf file

Dihedral groups: postscript file or pdf file

Symmetric groups: postscript file or pdf file

Alternating groups: postscript file or pdf file

Group examples: postscript file or pdf file

Rings and Modules: postscript file or pdf file

Rings and Modules, the proofs: postscript file or pdf file

Fields and Vector spaces: postscript file or pdf file

Fields and Vector spaces, the proofs: postscript file or pdf file

Office hours: Sunday 11:30-1:00pm in Van Vleck B239 and by appointment. For appointments, 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.

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 on Monday, in lecture. All claims that you make in your homework MUST BE PROVED in order to receive credit.

Homework assignments:

Homework assignment 1: DUE September 8, 2003 and September 15, 2003 postscript file or pdf file

Homework assignment 3: DUE October 13, 2003 postscript file or pdf file

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 (10 minute) vocabulary quiz each Friday during class.

Vocabulary lists:

Vocabulary list 1: postscript file or pdf file

Exams: There will be three 50 minute in-class midterms: October 1, October 22 and November 24. There will be a 2 hour final exam at 2:45pm Friday December 19. The exams will be a random selection of homework problems from the homework assignments. The final exam 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, drider@math.wisc.edu or in 821 Van Vleck. See the web page at http;//www.math.wisc.edu/~drider/FML.html

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.

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.