Right wrong turns

Sometimes it just all comes together, y'know? It helps having a class full of kids (and a few older folks) who are willing and able to put the responsibility for their learning on their own shoulders and run off with it, leaving me in the dust.

Several times during the past week I've found myself extemporizing in Abstract II as the students have asked questions the answers to which I had no idea. That's the satisfying/scary thing about teaching these high-end courses: on the one hand, the students are smart and savvy enough enough to ask really deep and interesting questions on the fly; on the other hand, the students are smart and savvy enough to ask really deep and interesting questions on the fly. It ain't Calc I in there: although I will never tire of the thrill of teaching calculus (Calc I folks: I'm not kidding when I say that I never get over the beauty of the definition of the derivative. That's not feigned excitement: I really am in awe of the limit definition, after all of these years, and I'll never get tired of seeing the hs magically disappear!), and though those first-year courses are often jam-packed with smart students, it's no more than once every few semesters that I find myself perplexed by a student's question in Calc I: there's just not enough that's unfamiliar, and the concepts are second nature to me by now.

But Abstract II? Wow. That class is a kaleidoscope, and whatever view we find on a given day seems to depend as much on the mood of the class as it does on the phase of the moon.

A couple of the questions that have come up lately in that course are pretty routine and standard ones I was able to respond to without skipping a beat: "If f(x) is irreducible in k₁[x] and k₂ is a subfield of k₁, then is f(x) irreducible in k₂[x]?" "What would an irreducible cubic polynomial in ℚ[x] look like?" (This last was in response to the proposition showing that the irreducible quadratic and cubic polynomials over any field k are precisely those with no roots in k.)

The really interesting stuff happened when I was momentarily fertumult.

"Why would you want to represent f(x) in 'base' b(x)?" asked Bertrand in class on Wednesday. He's always good for a truly probing question or two.

"Um..." For a moment I rethought my policy of encouraging students to ask good questions like why?

"Um...well...if we know that k is finite..." I hemmed and hawed for a few minutes and waved my arms around in an unconvincing fashion. A few muttered words seemed to make some sense, and reason returned, and I felt better as my mental wheels gained a bit of purchase. "...then if we've got a homomorphism φ:k[x] → k[x], φ is completely determined by its action on the finitely many polynomials of degree less than deg(b) and its action on b itself."

Whew!

I've not done so much tiptoeing as I've done with these kids since my first semester at the University of Illinois when they gave me a section of Accelerated Honors Calc III for engineers to teach...those kids were smart. It hadn't helped me that at the time I hadn't done vector integration since my second year of undergrad.

This past Monday's class was the funnest so far this week, and it came about because I'd been sloppy in setting up an example in my worksheet for that day. The exercise asked the students to come up with two quadratic polynomials in ℤ₈[x] which each exhibit more than one nontrivial factorization, thus demonstrating twice over that ℤ₈[x] is not a unique factorization domain.

One of my preplanned examples worked, the other was verkochte (I'd forgotten that in ℤ₈, -3 and 5 are identical, so the "distinct" factorizations I'd expected were one and the same). Did the students give up? No, no, no! Instead, after a few minutes of floundering about trying to fix the failed example I'd started with, we just set up the equation that would have to hold in order to yield two nontrivially distinct factorizations, and solved away. We soon had a whole passel (not a half of a passel, or even two-thirds of one) of examples.

That's a great class.

While I'm bragging on them for their self-directed successes, I should give props to Uri for coming up with the basis for a fantastic question for the first exam. Though several of the students, when asked last week as part of their homework to posit potential exam questions (with accompanying solution sketches), came up with appropriately difficult and interesting posers, it was Uri's question involving the advanced ring theoretic properties of the powerset ring that proved the most amenable to inclusion on the exam. It was just right.

Okay, off to read a few Project NExT-Southeast Fellow applications before I hit the hay.

Au revoir!