PlanetMath: criterion for constructibility of regular polygon

Proof. First of all, note that a primitive

th root of unity is a constructible number if and only if $\displaystyle \cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number. See the entry on roots of unity for more details. Therefore, without loss of generality, only the constructibility of the number $\displaystyle \cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ will be considered.

Sufficiency: If a regular -gon is constructible, then so is the angle whose vertex is the center of the polygon and whose rays pass through adjacent vertices of the polygon. The measure of this angle is $\displaystyle \frac{2\pi}{n}$ .

By the theorem on constructible angles, $\displaystyle \sin\left(\frac{2\pi}{n}\right)$ and $\displaystyle \cos\left(\frac{2\pi}{n}\right)$ are constructible numbers. Note that is also a constructible number. Thus, $\displaystyle \cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number.

Necessity: If $\displaystyle \omega=\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2\pi}{n}\right)$ is a constructible number, then so is $\omega^m$ for any integer .

On the complex plane, for every integer with $0\le m<n$ , construct the point corresponding to $\omega^m$ . Use line segments to connect the points corresponding to $\omega^m$ and $\omega^{m+1}$ for every integer with $0\le m<n$ . (Note that $\omega^0=1=\omega^n$ .) This forms a regular -gon. $\qedsymbol$