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

Math 541
Modern Algebra
A first course in abstract Algebra
Lecturer: Andrei Caldararu

Fall 2006

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 two one hour and 15 minute lectures per week:

TR 2:30-3:45 in B337 Van Vleck

Text: The textbook for the course is

Abstract Algebra, by David Dummit and Richard Foote, Third Edition, John Wiley and Sons, Inc., 2004.

Course Notes (largely due to Arun Ram):

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: Tuesdays and Thursdays 1:15-2:15 in Van Vleck 605 and by appointment.

Grading: The term grade will be based on homework and the exams as follows: Homework: 30% Midterm: 30% Final Exam: 40%. 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 a week after it is assigned, during lecture. All claims that you make in your homework MUST BE PROVED in order to receive credit.

Homework assignments:

Homework assignment 1: DUE September 21, 2006

Homework assignment 2: DUE October 10, 2006

Homework assignment 3: DUE October 26, 2006

Homework assignment 4: DUE November 16, 2006

Homework assignment 5: DUE December 5, 2006

Homework assignment 6: DUE December 18, 2006 in my mailbox on second floor of Van Vleck Hall. Solutions.

Exams: There will be one 75 minute in-class midterm on October 26. There will be a 2 hour final exam at 10:05am Wednesday, December 20, in room B113, Van Vleck Hall. The exams will be a random selection of homework problems from the homework assignments. The final exam will be cumulative.

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.