Trevor Hyde ’12
Major: Mathematics
Thesis Advisor: David Cox

What is your thesis about?

The thesis is about the Galois Theory of division points in the lemniscate. The lemniscates is a curve that looks like a figure-eight, and it’s given by a polar equation, r2=cos2θ. I’m looking at points that divide this curve into equal arc-length segments and studying the symmetries that you get from creating a new number system where you incorporate these points. The symmetries turn out to have lots of really interesting properties. What I did in my thesis was compute what are called the Galois groups, which measure the symmetry of these number systems, and that hadn’t been done before. That allowed me to give a new proof of a theorem of Abel’s called Abel’s theorem on constructing division points of the lemniscate. He characterized the points on this curve: the division points are the points on this curve into equal arc-length segments. If you think of this curve living in the plane, then you could ask, if you really wanted to, which of these points could you construct with a ruler and compass. Greek geometers used to construct lots of things with a ruler and compass, and one thing they liked to do was create regular polygons. A regular polygon is a polygon where all of the sides have the same side length. Gauss showed that constructing regular polygons is the same thing as constructing division points of the circle, so if you want to be able to make, say, a pentagon, you needed to be able to divide the circle into five equal arc-length segments. And then Gauss was able to show, using Galois Theory, that, rather amazingly, you can do this if and only if n can be written as a power of two times a product of distinct, what are called Fermat, primes. In his book, “Disquisitiones Arithmeticae,” he alludes to the fact that this theory could be extended more generally, but he doesn’t say how. Abel picked up on this, and he figured out that one of the ways you could extend it was to this curve, the lemniscates, and if you talk about the division points on here, then you can construct these points, which are sort of a weirdly-shaped regular polygon that don’t really look like polygons. You can construct these with a ruler and compass if and only if n is exactly of the same form [as in the case of a circle]. So it gave a characterization of these points, and it also uses Galois Theory, however the proof was much harder, and it took a lot more work. One of the highlights of my thesis is giving a brand-new, concise proof of this theorem.

How did you come up with this thesis topic?

Well, I always knew I liked number theory, so this lived in number theory. But also I knew I wanted to do something where a lot of different branches of math were coming together. I like the interface of different mathematical disciplines. So I proposed that idea to Professor Cox, who’s my advisor. At the time I was doing a special topics course in Galois Theory, and at the very end of the course and at the very end of his book on Galois Theory, there’s a chapter on this lemniscate curve, where you prove this theorem using Galois Theory. The proof is somewhat unsatisfying, not very elegant. And it’s long and it’s very hard, and so I wanted a nicer proof. I wanted a proof that was more like the one for the circle, because that’s a very elegant proof. So I embarked upon learning what’s called class field theory, because Professor Cox thought that that might work to help me solve this. I learned a lot of class field theory, which I also incorporated into my thesis, but it turned out that I didn’t need it to prove the theorem. At the very end of last semester, I found a proof in a completely different direction of the theorem.

What advice would you have for others considering writing a thesis?

Well, I would say let yourself get really involved in the topic, sort of immerse yourself in the subject. I definitely made it the thing I was thinking about all the time, and so it was the focus of my thought for most of last semester and this semester, as well. It was really fun and rewarding to get really, really in to one specific subject. And I think just all of that time and effort led me to coming up with my result. So that’s my advice: Don’t put it off until the second semester.

You’re going into a career in math?

Hopefully. My goal is to be a researcher and a teacher, so it looks like professor is what I’d like to do. I’m taking next year off to do … I don’t know yet, but then I plan on going to graduate school the year after that.

Does writing a thesis help you in getting into grad school?

I don’t know if it will help me get into grad school. It definitely helped me figure out if I wanted to be a researcher. The experiences I had before were all research projects that someone else designed for me, you know, Research Experience for Undergraduates or things like that. And while they were interesting and good experiences, I didn’t have the same passion for it because it wasn’t my project, necessarily. So in this, I really got to decide what I was going to be doing, and how I would do it, and it was only me doing it. I definitely felt ownership of the project and that helped me have a lot more fun with it, and I realized that that would be something I would like to do with my time.