\section{The Constructible Regular $\mathbf{n}$-gons}

\begin{lemma}
	Let $n$ be a positive integer and let $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$. Then $\Q(\cos(2\pi/n)) \subseteq \Q(\omega)$.
\end{lemma}

\begin{theorem}[(Gauss, 1796)]
	It is possible to construct the regular $n$-gon with a straightedge and compass if and only if $n$ has the form $2^kp_1p_2\dots p_t$, where $k \geq 0$ and the $p_i$'s are distinct primes of the form $2^m + 1$.
\end{theorem}