Math 441 is designed to serve (at least) two purposes: (i) It can help you do well in Math 541, if you are not yet confident about your ability to construct proofs, and (ii) it is a significant component in the program preparing secondary mathematics teachers. Our course is called “Introduction to Modern Algebra” while 541 is called “Modern Algebra”, which makes some sense of (i). But why does this material belong in the preparation of a secondary math teachers? Many students in high school mathematics courses can blame some of the difficulty they find in those courses on not clearly understanding what came earlier: arithmetic, yes, but why does arithmetic work the way it does rather than just the ability to imitate a calculator. (Do you recognize yourself here?) Quite a bit of 441 can be described as just that understanding of how our numbers work.
Over the last few centuries there have been a lot of cases where people came really to understand some part of mathematics by throwing away what was inessential: Is our arithmetic really about numbers? Or what parts of it might make just as much sense applied to other things. Most of you have already seen this in Math 340, where the mathematics of matrices was shown to behave a lot like arithmetic. But not exactly like ordinary arithmetic (e.g. you can find non-zero things whose product is zero, and you can’t do that with our usual numbers) so what determines which things carry over and which do not? In 441 we will consider both facts about arithmetic and some other generalizations to non-numbers. Understanding the generalizations casts a lot of light on arithmetic.
But now you are very fluent with properties of arithmetic, so why do you need this understanding? For those of you preparing to be teachers, one reason it is really critical to understand not just how arithmetic or algebra works but why is this: Suppose you are a teacher and some day a student comes up and says “I did this problem a different way from the book’s way, but I got the same answer. Is that OK?” Regardless of whether there is some reason that your students need eventually to be able to use the standard ways of doing the problem, you need to be able to see whether that student’s way would work for all problems like this, whether it only works because of the particular numbers involved, whether if it always works it is in some circumstances going to be much harder than the standard way, etc. And students can, and do, come up with ways that are very unexpected so you can’t possibly just memorize all the ways they might try. But even if you are not going to be answering questions like that, “generalized arithmetic” is important in many modern applications. Things like error-correcting codes (essential in computing and data transmission) and cryptography, as well as many other tools, can be direct applications of parts of modern algebra.
We will talk about systems which look a lot like the ordinary integers but are not made up of numbers. Some of those systems are called rings, some are called groups, some are called fields, etc. Since it happens that the integers together with our usual rules for addition and multiplication are a ring, anything we learn about rings will be true for the integers, without depending on properties somehow related to being numbers. And there are rings other than the integers that have real applications such as the coding and en/decryption referred to above, and anything we learn about rings will apply there as well as to the integers. We will be studying how these abstractions from ordinary arithmetic work. Along the way you will recognize some of the things you have seen in other courses called algebra, e.g. polynomials. The Childs’ textbook includes sections on many applications. We won’t have time to do many of those in detail, but do feel free to ask me about them.
The official prerequisite for Math 441 is Math 340. We won’t be doing much matrix calculation but we will fairly frequently refer to sets of matrices and their properties as examples for what we do learn. Likewise, 340 implies that you have been through either math 234 or 222 and also 240: We won’t be doing much with calculus except that again it gives us some useful examples.
We will have two “midterm exams” and a final exam, as shown on the course schedule at our class website http://www.math.wisc.edu/~wilson/Courses/Math441/Spring2010. Our final exam is scheduled for 7:45 AM on Monday, May 10. That is very early in the exam period so it should not cause any trouble with travel plans. I will also be assigning homework problems from the book. I will make a long list of problems that you ought to make sure you can do, as a test for yourself of how well you understand the material, but I will select only certain ones to be written up and handed in. Final grades will be based on the exams and homework: I hope to be able to include some group-work activities during the semester, and will work out a way to factor those into the final grade also. To begin with you can think of grades as coming from a total of 500 possible points: 200 for the final, 100 for each midterm, and 100 for the homework-to-turn-in assignments. That may be slightly modified if we do include some group-work.
Bob Wilson