Of course, the image above omits the seventh axiom for a ring: For all a,b,c in R, one must have a(b+c)=ab+ac, and (b+c)a = ba+ca (the distributive law for addition). To the right of the definition are addition and multiplication tables for Z_7, and the multiplication table for Z_8.

Course information

This course will explore abstract algebraic structures related to integers, polynomials, and the fields of rational, real, and complex numbers. As a consequence of this exploration, we will be able to answer questions that seemingly have little to do with algebra. For example, we will show that it is impossible using only a straightedge and compass to double the cube, square the circle, or trisect an arbitrary angle. A main goal of this course is to show that abstraction, properly motivated by examples, can lead to greater understanding.

In this course you will increase your mathematics content knowledge and improve your ability to express mathematical ideas in writing. My role as the instructor will be to provide information and guidance, to set expectations, and to assess how well students meet those expectations. Your role as a student will be to work hard, to read the text, to participate in class activities, and to work on out-of-class assignments, including assignments that may not be graded.

In this course you will be assessed not only on your problem solving skills but on your ability to clearly explain the material. Solving the types of exercises we will consider, and writing clear, complete solutions to those exercises, will require focus, attention, and effort every day. Doing well in this course will require a lot of work, but the results will be worth it. If you're having trouble reading the textbook, doing the homework, or understanding any of the course material, please come see me right away. I'm here to help.

Expected learning outcomes for Math 4000/6000.

Homework

Other material

Here are some links to additional material that you may find useful or relevant to this course.

LaTeX

Though you are not required to type your homework for this class, I encourage you to do so. Typed homework is much easier for me to read, and I think that students often take their work more seriously when they type it than when they write it out by hand. One option for typing your work is to use Microsoft Word, and to input mathematical expressions using Word's built-in equation editor. My preferred choice is LaTeX (pronounced as either LAY-tek or LAH-tek). LaTeX is a free, open-source mathematics typesetting program that is the standard for most professional mathematics writing. Writing mathematics with LaTeX is a lot like coding a web page in HTML. Both tasks require using special syntax, and both tasks require a special program to produce viewable results.