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

Constructing a regular $n$-gon with straightedge and compass, inscribed in a given circle, is equivalent to constructing the central angle $2\pi/n$, which is equivalent to constructing $\cos(2\pi/n)$, or in complex-number terms, the primitive $n$-th root of unity $\zeta_n = e^{2\pi i/n}$. Ancient constructions handled $n=3,4,5,6,8,10,12,15,\ldots$ (triangle, square, pentagon, and their angle-bisected and combined variants), but no fundamentally new odd case was found for two millennia.

Carl Friedrich Gauss broke this in 1796, publishing full details in Disquisitiones Arithmeticae (1801, §VII). For $n=17$, the relevant algebraic quantity $\zeta_{17}$ satisfies the cyclotomic equation $1+\zeta_{17}+\zeta_{17}^2+\cdots+\zeta_{17}^{16}=0$, a degree-$16$ equation. Gauss noticed $16=2^4$ and, crucially, that $17-1=16$ is itself a power of $2$ — this numerical coincidence is exactly what makes the polygon constructible, as later steps make precise.

Gauss's method (Steps 2–4) groups the $16$ nontrivial $17$-th roots of unity into nested sums called periods, each satisfying a quadratic equation over the previous stage; the surviving steps (5–6) show why this pattern — and hence constructibility — depends only on whether $\varphi(n)$, Euler's totient of $n$, is a power of $2$.