Worked solution: The Gauss–Wantzel theorem: constructible regular polygons via cyclotomic fields

Gauss's 1796–1801 work established sufficiency: whenever $\varphi(n)$ is a power of $2$, the regular $n$-gon is constructible. He asserted, without a published proof, that the condition was also necessary. Wantzel supplied this missing direction in the same 1837 paper that settled angle trisection and doubling the cube, reusing his general criterion that a constructible real number has degree a power of $2$ over $\mathbb{Q}$.

The key observation is that $\zeta_n$ satisfies the quadratic equation $\zeta_n^2 - 2\cos(2\pi/n)\,\zeta_n + 1 = 0$ over the field $\mathbb{Q}(\cos(2\pi/n))$ (obtained from $\zeta_n + \zeta_n^{-1} = 2\cos(2\pi/n)$ and $\zeta_n\zeta_n^{-1}=1$), so $[\mathbb{Q}(\zeta_n):\mathbb{Q}(\cos(2\pi/n))] = 2$ exactly (it cannot be $1$ since $\zeta_n$ is not real, but $\cos(2\pi/n)$ is). By the tower law, $\varphi(n) = [\mathbb{Q}(\zeta_n):\mathbb{Q}] = 2\cdot[\mathbb{Q}(\cos(2\pi/n)):\mathbb{Q}]$, so the constructible real length $\cos(2\pi/n)$ has degree $\varphi(n)/2$ over $\mathbb{Q}$.

By Wantzel's general constructibility criterion, this degree must be a power of $2$ for $\cos(2\pi/n)$ to be constructible — which forces $\varphi(n)$ itself to be a power of $2$. Combined with Gauss's sufficiency, this completes the Gauss–Wantzel theorem: the regular $n$-gon is constructible with straightedge and compass if and only if $n=2^k p_1\cdots p_m$ for distinct Fermat primes $p_i$ (Wantzel 1837, §IV).