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The Big Picture of abstract algebra?

July 2, 2007, 1:17 pm

 English Images FarmarialI’ve been working here and there on my Modern Algebra class for this fall. As regular readers know, I am doing this course this time around without the use of a required textbook. One of the difficult, and good, things this approach imposes on me as the professor is that I cannot rely on the book to provide structure and order to the course. I have to do this myself. Before I can do any realistic planning, I first have to decide what I am going to cover and the order in which I am going to (try to) do it. And before I can do that, I have to face some questions that professors are surprisingly able to sidestep when using a textbook, namely: What is this course about? What themes unify, and therefore motivate, the material? And what are the core issues and questions that this course attempts to address?

Far too often, students can take a course in college or high school and make good grades, and even are able to do some of the tasks that the course outcomes in the syllabus require. But they are not able to explain at the end of the semester or school year just exactly what it is that they just studied. They are deprived of the Big Picture, the aerial shot that shows where everything is in context, because nobody ever thought to bring it up. They and their teachers just brachiate from one topic or calculation to the next without thinking of where it all fits and what it’s all about. Whether this is out of the teacher’s sheer laziness, incompetence in the subject matter, or just not having enough time to do it, is irrelevant — students can’t be expected to demonstrate real mastery of a subject if they don’t know what it’s about and what it was invented for. If they have that mastery in the absence of the Big Picture, it’s a coincidence.

That strongly worded statement out of the way, here is my initial attempt at articulating the big issues and questions in Modern Algebra.

Students in this course have, by the first day of class, completed a gamut of advanced mathematics classes including two semesters of calculus, a course in advanced problem solving, and a semester of linear algebra. In those courses, and going back to high school, we’ve seen all kinds of things one can do with mathematics. All of the things we do in math tend to revolve around certain kinds of objects that we do things to. Perhaps the six most prominent types of objects we encounter in math are

  • The integers
  • The rational numbers
  • The real numbers
  • The complex numbers
  • Matrices
  • Polynomials

And as far as doing things to these objects, there is — somewhat surprisingly — a lot of overlap. In particular, we have a notion of addition for each of these and a notion of multiplication. Although the specifics differ — multiplication of 2×2 matrices is a lot different than multiplying two polynomials, for instance — these two operations give us many things to do with all six of these objects. And high school algebra is spent pushing these operations to their limits, doing things like solving equations and factoring and so on.

The purpose of Modern Algebra is to find out what we can do with these sets and the operations of multiplication and addition, and — crucially — any other set of objects with similar operations that behave like these sets. Throughout the course, one takes the Big Six sets of objects and any other set with operations that behaves in some sense (to be defined later) like them — for example, matrices with only integer entries, or fractions whose denominators are powers of 2, or polynomials with matrix coefficients — and asks the following questions. These questions are the essential motivating themes of the course:

  1. What sorts of arithmetic properties hold? For example, do we have the distributive property, the commutative property for one or both operations, etc.?
  2. What kinds of arithmetic properties do all of our sets with operations have in common? Which ones come close?
  3. Can we solve equations with these objects and their operations?
  4. Is there a sort of “basis” for each set with operations, in the sense that any element in the set can be built out of “basis” elements using the operations provided? [Example: Prime numbers form a "basis" for the integers if you use multiplication (and include 0 and -1).]
  5. If so, then is it possible to take an element of our set and factor it into a combination of “basis” elements?
  6. If so, then is that factorization unique?

In other words, the whole course comes down to four basic problems:

  • Defining arithmetic and deriving arithmetic properties (or failing to do so, and taking note)
  • Equation-solving
  • Finding a “basis” for a set with operations
  • Factoring, with a view toward unique factorization (or determining that factorizations are not unique and examining the consequences)

I think those six questions and the four basic problems provide a nice umbrella for the entire course — which will run through basic group theory and a bit of rings and fields — in a way that definitely “feels like algebra” for the students. I’m still working on how best to run the day-to-day classes and how best to assess students, but I believe that every day and in everything students do, we’ll come back explicitly to these two lists.

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