Lecture 5
Ruler and Compass Constructions II
 

5. Constructible regular polygons


In this section we discuss constructibility by ruler and compass of regular polygons. Gauss' proved that a regular polygon of $ n$ sides is constructible by ruler and compass if and only if $ n=2^m p_1p_2 \ldots p_r$ where $ m \in \mathbb{N}$ and $ p_1,p_2, \ldots,p_r$ are distinct Fermat primes. The number $ F_{m} = 2^{2^{m}} +1$ is called a Fermat prime whenever it is a prime. The known Fermat primes are:

$ F_{0}=3, F_{1}=5, F_{2}=17, F_{3}=257$ and $ F_{4}=65537.$

Fermat showed that $ F_{m}$ is a prime for $ m \leq 4$. Eisenstein conjectured that there are infinitely many Fermat primes. This conjecture is still open. Euler showed that $ F_{5}$ is divisible by $ 641$.


Proposition 5.1

(1837, Wantzel)   Let a regular polygon of $ n$ sides be constructible and $ p$ be an odd prime dividing $ n$. Then $ p$ is a Fermat prime.


Proof.
If $ p\vert n$ and a regular $ n$-gon is constructible then a regular $ p$-gon is also constructible. Thus the point $ (\cos2 \pi /p, \sin 2 \pi/p)$ is a constructible point. Hence there exists a field $ F \supseteq \mathbb{Q}$ such that $ [F:\mathbb{Q}] =2^{m}$ and $ \cos2 \pi /p, \sin2 \pi /p \in F$. Then $ \zeta_p = \cos2 \pi /p + i \sin 2 \pi /p \in F(i)$ and hence $ [\mathbb{Q}(\zeta_p) : \mathbb{Q}] = 2^{s} = p-1$ and therefore $ p=1+2^s.$ It follows that $ s$ is a power of $ 2.$ Hence $ p$ is a Fermat prime.

Figure : Richmond's construction of a regular pentagon