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

Frequently asked questions

Does every constructible number have a formula this explicit?

In principle yes: the same tower-of-square-roots argument that produces this formula for $\cos(2\pi/17)$ works for any constructible number, though the formula can become far longer as the tower gets taller. What is special about $\cos(2\pi/17)$ is that it was the first genuinely new case discovered since antiquity.

Source

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

Disquisitiones Arithmeticae

Step 4 of 8: The payoff: an explicit nested-radical formula

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In plain words

Following the tower down from $\mathbb{Q}(\zeta_{17})$ to $\mathbb{Q}$ step by step, and solving each quadratic equation along the way with the ordinary quadratic formula, eventually spits out an explicit expression for $\cos(2\pi/17)$ made only of whole numbers, additions, and nested square roots. It looks intimidating, but every single square root inside it is something a compass can draw directly.

This formula is the concrete, checkable proof that the abstract field-theory argument was not just a trick — climb down the tower of quadratics, and out comes an honest number you could, in principle, mark on paper.

\cos\frac{2\pi}{17}=\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}\right)

Detailed analysis

Solving the tower of quadratics from Step 3 explicitly — expressing each period in terms of the previous stage via the quadratic formula, and substituting all the way down to $\mathbb{Q}$ — Gauss obtained the closed form $\cos\frac{2\pi}{17}=\frac{1}{16}\left(-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+2\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}\right)$ (Gauss 1801, Disquisitiones Arithmeticae, Art. 365). Every operation used — addition, subtraction, multiplication, division, and square roots of already-constructed lengths — corresponds directly to a straightedge-and-compass step, so the formula itself is a certificate of constructibility.

This is not merely a numerical curiosity: it is the payoff of the entire chain of reasoning in Steps 1–3, made concrete enough to check by direct substitution. In 1893, Herbert Richmond translated the algebra back into an explicit geometric construction with ruler and compass, giving a step-by-step picture that realizes this formula.

Having demonstrated sufficiency for $n=17$ by explicit construction, the remaining steps generalize Gauss's argument: which values of $n$ admit a tower of square-root extensions like this one, and (via Wantzel) why no other values do.

Terms in this step

Closed-form (radical) expression: A formula for a number built only from integers and the operations $+,-,\times,\div$ and taking $n$ -th roots — no infinite sums or unnamed limits — so that it can, in principle, be evaluated by hand or built geometrically.

Knowledge used in this step

Group theory