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

Frequently asked questions

Did Gauss actually compute every one of these quadratic equations by hand?

Yes — Gauss carried out the full computation in Disquisitiones Arithmeticae (1801), eventually deriving an explicit nested-radical formula for $\cos(2\pi/17)$ , shown in the next step. A geometric ruler-and-compass construction realizing these steps was later given explicitly by Herbert Richmond in 1893.

Source

Carl Friedrich Gauss (sufficiency, 1796/1801); Pierre Laurent Wantzel (necessity, 1837), 1837

Disquisitiones Arithmeticae

Step 3 of 8: The pattern repeats:

8

-periods,

4

-periods,

2

-periods, then the roots

Previous step Next step

In plain words

Having split the $16$ roots into two groups of $8$ , Gauss simply repeats the trick: split each $8$ -group into two $4$ -groups, whose sum and product can again be computed from the previous stage's numbers, giving more ordinary quadratic equations. Then split each $4$ -group into two $2$ -groups the same way. Finally, split each $2$ -group into its two individual roots.

Every single split is a quadratic — a square root — because the doubling structure of the exponents (powers of $3$ ) keeps working at every scale. Four rounds of halving turn $16$ roots into $1$ , so the whole tower uses exactly four square roots stacked on top of each other.

\mathbb{Q} = F_0 \subset F_1 \subset F_2 \subset F_3 \subset F_4 = \mathbb{Q}(\zeta_{17}), \quad [F_i:F_{i-1}]=2

Detailed analysis

The same sum-and-product argument used in Step 2 repeats at every scale. Splitting $\eta_{1,0}$ and $\eta_{1,1}$ (each a sum of $8$ roots) into four periods of $4$ terms gives sums and products expressible in terms of $\eta_{1,0},\eta_{1,1}$ and rational numbers, so each pair of $4$ -term periods satisfies a quadratic equation over $\mathbb{Q}(\eta_{1,0})$ . Splitting those $4$ -term periods into $2$ -term periods repeats the argument once more, and splitting the $2$ -term periods (such as $\zeta+\zeta^{16}=2\cos(2\pi/17)$ ) into the individual roots $\zeta$ and $\zeta^{16}$ closes the tower (Gauss 1801, Disquisitiones Arithmeticae, Art. 354–361).

This produces a tower of fields $\mathbb{Q}=F_0\subset F_1\subset F_2\subset F_3\subset F_4=\mathbb{Q}(\zeta_{17})$ where each step adjoins one square root, so $[F_i:F_{i-1}]=2$ for $i=1,2,3,4$ . By the tower law, $[\mathbb{Q}(\zeta_{17}):\mathbb{Q}]=2^4=16$ , matching the degree of the cyclotomic equation from Step 1 — a consistency check that the construction is complete and uses no extra steps.

Because every stage of the tower is a quadratic extension, and quadratic extensions correspond exactly to straightedge-and-compass steps (as in the general theory used for angle trisection and doubling the cube), $\cos(2\pi/17)$ — and hence the regular $17$ -gon — is constructible.