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

Constructing a regular $n$-gon inscribed in a given circle with straightedge and compass is equivalent to dividing the circle into $n$ equal arcs — that is, constructing the central angle $2\pi/n$, or in coordinates, the real length $\cos(2\pi/n)$. In the complex plane this means constructing the primitive root of unity $\zeta_n = e^{2\pi i/n}$. Euclid's Elements already showed how to construct the regular triangle, square, and pentagon ($n = 3, 4, 5$), and how to double the number of sides by angle bisection ($6, 8, 10, 12, 16, \ldots$), as well as combining coprime cases ($n = 15$ from $3$ and $5$). For over two thousand years, no new odd prime was added to that list.

On 30 March 1796, nineteen-year-old Carl Friedrich Gauss discovered that the regular $17$-gon is constructible — the first new constructible regular polygon with an odd number of sides since antiquity. The result appeared in §VII of his Disquisitiones Arithmeticae (1801). The discovery did not proceed by geometric trial: Gauss translated the problem into the algebra of the cyclotomic equation $x^{16} + x^{15} + \cdots + x + 1 = 0$, whose degree is $\varphi(17) = 16 = 2^4$.

The numerical coincidence $17 - 1 = 16 = 2^4$ is the pivot of the whole discovery: because $16$ is a power of $2$, the $16$ roots can be partitioned into successive halves — two periods of $8$, then four of $4$, then eight of $2$ — each stage satisfying a quadratic equation with coefficients determined by the previous stage. The remaining steps trace this hierarchy from Gauss's explicit formula to the general Gauss–Wantzel criterion for every $n$.