For which positive integers \(n\) is a regular \(n\)-gon constructible by ruler and compass?
Solution
By the Gauss–Wantzel theorem, a regular \(n\)-gon is constructible if and only if \[ n=2^k p_1p_2\cdots p_r, \] where \(k\ge0\) and the \(p_i\) are distinct Fermat primes, \[ p_i=2^{2^{m_i}}+1. \]
Indeed, a regular \(n\)-gon is constructible exactly when the primitive \(n\)th root of unity \(\zeta_n\) is constructible over \(\QQ\). This is equivalent to the degree \[ [\QQ(\zeta_n):\QQ]=\varphi(n) \] being a power of \(2\) and the cyclotomic extension having a tower of quadratic subextensions. The condition that \(\varphi(n)\) be a power of \(2\) is equivalent to the displayed factorization of \(n\).
Thus powers of \(2\) are always allowed, and every odd prime factor must be a distinct Fermat prime.