criterion for constructibility of regular polygon

Theorem 1.

Let $n$ be an integer with $n\mathrm{\ge }\mathrm{3}$. Then a regular $n$-gon (http://planetmath.org/RegularPolygon) is constructible (http://planetmath.org/Constructible2) if and only if a primitive $n$th root of unity (http://planetmath.org/PrimitiveRootOfUnity) is a constructible number.

Proof.

First of all, note that a is a constructible number if and only if $\cos \left({\displaystyle \frac{2\pi }{n}}\right)+i\sin \left({\displaystyle \frac{2\pi }{n}}\right)$ 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 $\cos \left({\displaystyle \frac{2\pi }{n}}\right)+i\sin \left({\displaystyle \frac{2\pi }{n}}\right)$ will be considered.

Sufficiency: If a regular $n$-gon is constructible, then so is the angle whose vertex (http://planetmath.org/Vertex5) is the center (http://planetmath.org/Center9) of the polygon and whose rays pass through adjacent vertices of the polygon. The measure (http://planetmath.org/AngleMeasure) of this angle is ${\displaystyle \frac{2\pi }{n}}$.

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

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

On the complex plane, for every integer $m$ with , 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 $m$ with . (Note that ${\omega }^0=1={\omega }^n$.) This forms a regular $n$-gon.∎