Cutting curves into equal parts

The ancients were concerned with what lengths could be constructed from a given length by means of certain operations with an unmarked straight edge and a compass. In modern language, a number $\alpha\in \mathbb{C}$ can be constructed if and only if $\mathbb{Q}(\alpha)$ is a subfield of a field $K/\mathbb{Q}$ for which there is a tower $$\mathbb{Q}=K_0\subset K_1 \subset K_2\subset \dots \subset K_n=K$$ with each extension $K_{j+1}/K_j$ of degree 2.

Gauss proved that the cyclotomic number $e^{2\pi i/n}$ is constructible if and only if $n$ is a power of 2 times a product of distinct Fermat primes (i.e. primes of the form $2^n+1$). This answers the question of whether it is possible to divide a circle into $n$ equal parts using compass and straight edge. In the article below, Michael Rosen gives a modern review of Gauss' theorem and then goes through a less well known theorem of Abel about the lemniscate, which is the figure 8 curve satisfying $$(x^2+y^2)^2=x^2-y^2.$$ Abel proved that it is possible to use ruler and compass to divide the lemniscate into $n$ equal length parts if and only if $n$ is a power of 2 times a product of distinct Fermat primes.

Michael Rosen
Abel's Theorem on the Lemniscate
The American Mathematical Monthly, June, 1981, pp. 387-395
Online article here

Go through the proof of Abel's theorem and learn the Galois theory of elliptic curves.

Another possible project would be to look at the generalization of Abel's theorems to ``clovers'' by Cox and Shurman. A lemniscate is something like a ``two-leaved clover''. Generalizations of Abel's theorem to multi-leaved clovers are proved in the recent Monthly paper:

David A. Cox and Jerry Shurman
Geometry and Number Theory on Clovers
The American Mathematical Monthly, October, 2005, pp. 682-704
Excerpt of article available here

This is also an excellent paper to learn some of these Galois theory problems.

Another interesting project would be to study Gleason's paper on a variation of Gauss' theorem. Gauss showed the regular 13-gon (``triskaidecagon'') cannot be constructed by straight edge and compass. However, suppose we allow certain other operations, such as the ability to trisect angles. Then what regular polygons may we construct?

Title: Angle Trisection, the Heptagon, and the Triskaidecagon
Authors: Andrew M. Gleason
Source: The American Mathematical Monthly, Vol. 95, No. 3. (Mar., 1988), pp. 185-194.
Online article here