It's such a perfect day

Well...perfect might be a stretch, but it was a good one, capping off a great week. The jewel in the crown of today's classroom antics shone brightly in the afternoon sun.

As soon as I reached campus at 8:30 I began fielding questions regarding the second of two homework problems I'd assigned to my Abstract II students for today; Quincy caught me as I walked in the door to the department with a couple of questions. A few hours later he was at it again in the Math Lab, and Derrick, Nadia, Hermann, Opal, and Katya all had difficulties too. Clearly the question (to which I openly admitted I knew not all the answers) was proving rougher than I'd thought it would.

The problem is simple to state: starting with the field of rationals, describe the various intercontainments of the following sorts of extensions that field: all extensions, finite extensions, algebraic extensions, normal extensions, pure extensions, and radical extensions. (Each containment or failure of containment had to be justified by proof or example.)

Between five or six of us, by 2:30 (class begins at 2:45) we'd catalogued examples and proofs verifying all but a couple of the containments/noncontainments we needed to show, but it was clear that the best course of action was to bring it up together in class.

That we did, and our team solution was an exciting one.

Several of the containments were easy and indisputable: pure extensions are radical, radical extensions are finite, finite extensions and algebraic extensions are one and the same. What wasn't clear was how normal extensions fit into the picture. It took us about half an hour of field wrangling before we'd found all of the examples we'd needed, constructing a convoluted splitting field here, a two-step radical tower there.

We were done.

Or so most of us thought.

Miguel had the temerity to ask about the exact location of the complex numbers in the scheme we'd laid out.

After a Moonlightingesque (kids, ask your parents) three minutes of high-level mathematical crosstalk, we remembered that the equivalence of two characterizations of normal extensions only holds for finite extensions (of which the complex numbers are not one), and therefore we were definitely allowed to call the complex numbers a normal extension of the rationals.

I realize that much of the above discussion will be over the heads of uninitiated, but I wanted to provide something close to a blow-by-blow account in order to highlight the phenomenal work of several of my students. Derrick, Nadia, Quincy, Miguel, Hermann, and Bertrand all contributed substantially to the conversation. The result was a true work of social mathematics, a finely-crafted piece of collaborative art.

Moreover, through our class today the students and I all gained a better understanding of the interrelationships of the manifold definitions we've encountered over the past few weeks; I can think of few fifty-minute periods this past semester that have been more mathematically or pedagogically satisfying. Though the exercise was definitely more challenging than I had intended it to be, it proved to be not insurmountably challenging, and the result of the solution was to solidify our fundamental understanding of several tricky concepts. This exercise is definitely a keeper, and I may just use it as a classroom activity that next time I teach this course (however long from now that will be).

I wonder if I might find similar exercises for my upcoming classes next fall? Or for the REU students this summer as they undertake a brutal and bruising survey of graph theory in the second week of June...?