Syllabus

Since I am teaching this course for the first time, I won't really know what the syllabus is until after the course is over. I understand the difficulties that causes for you, and will try to keep this page up to date - filling in what we have covered so far, and, when possible, where I think we will be going next. I may rely on people from the class to provide notes from time to time.

Topics
- Abstract algebra, culminating in proofs about the constructability (or not), with straightedge and compass, of various geometric figures.
- Elementary number theory.
Metatopics
- What makes mathematics interesting (or not)?
- How can our study of (relatively) advanced mathematics make a difference in how you teach less advanced material in high school?

January 24. Introductions. Mathematical autobiographies. The Euclidean algorithm and the fundamental theorem of arithmetic.
January 31. Feedback based on first week's reading and first hw assignment. Euclidean algorithm (concluded). AbstractionL rings and fields. The "usual rules of arithmetic" (Chapter 2 of Sethuraman and various places in Courant and Robbins).
February 7. Class starts a few minutes late ... New material: standard names for rings Z, Q, R, C, Z_n. R[x] is the ring of polynomials in the variable x with coefficicents from the ring R. Subrings. Additive cancellation is a consequence of the existence of additive inverses (you can always find a unique (-a) but multiplicative cancellation is trickier. Sometimes you can do it even when you don't have multiplicative inverses - for example, in Z 3a=3b implies a-b even though you don't have (1/3) to multiply by. So we have some definitions. An integral domain is a ring with no zero divisors - that is, a ring in which ab=0 implies a=0 or b=0. The integers are an integral domain. In an integral domain you have the cancellation law for multiplication. A field is a ring in which every nonzero element has a multiplicative inverse. We write (1/r) for the multiplicative inverse of r. Thus in Z₇ (1/2) = 4 since 4*2=1.
February 14. Field extensions. Vector spaces - starting into Chapter 3. We need to understand the definitions at the beginning of the chapter, the examples, and the theorem that establishes the idea of dimension. But we won't try to read the proof of that theorem. So you are responsible (roughly) for the material through page 70, definition 3.10 (telling what a basis is) and Corollary 3.17 and Definition 3.18).
Well, that's what I intended. What we actually did was spend about 2/3 of the time on the irrationality of sqrt(2) and on rational approximations to it.
February 21. Go over homework. Formal manipulations of polynomials and power series (as opposed to viewing them as functions). Vector spaces. Proofs from axioms. Dimension. (Material in this syllabus from last week that we never got to.)
February 28. Go over homework. Plans for Sethuraman material. Rational, algebraic and transcendental numbers. Infinite sets. Countability and uncountability. Hilbert's hotel.
March 7. First half: review hw, show that there are countably many algebraic numbers and uncountably many reals, hence "most" numbers are transcendental. Important digression into representing numbers using bases other than 10, in order to be able to use binary decimals to match the set of subsets of {1,2,...} (the committees of mathematicians, in the Hilbert Hotel model) with the infinite (binary) decimals representing numbers in the unit interval. Second half: noneuclidean geometry, in order to discuss the notion of independence for a set of axioms. Spent a fair amount of time on the Poincare model of the hyperbolic plane (more than one parallel to a line exists through a point not on the line). I got the main ideas right, and the notion of distance absolutely wrong (as I realized on the way home). I googled mathworld.wolfram.com/PoincareHyperbolicDisk.html. There's a cool on line program at www.math.umn.edu/~garrett/a02/H2.html that allows you to "draw" figures in this model and drag them around.
If you want more adventure, google elliptic geometry and noneuclidean geometry.
I'd really hoped to get as far as consistency (as well as independence) in order to be able to talk about the independence and the consistency of the continuum hypothesis ( is there an infinity strictly between the size of the integers (aleph-0) and the size of the real numbers (c, which is 2^{aleph-0)? Of course I never got
there.}
Passed out a selection of books for a spring break book report. <>
March 11-19. Spring break.
March 21.
March 28.
April 4.
April 6. Pass/fail, withdraw deadline.
April 11.
April 18.
April 25.
May 2.
May 9. Last class.