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

Frequently asked questions

So exactly how many regular polygons with an odd number of sides are known to be constructible?

With only $5$ known Fermat primes, there are exactly $2^5-1=31$ nonempty subsets of them, giving $31$ odd values of $n$ (products of distinct Fermat primes) for which the regular $n$ -gon is known to be constructible — plus all their doublings $2n, 4n, 8n,\ldots$ by repeated angle bisection.

Source

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

Disquisitiones Arithmeticae

Step 8 of 8: Legacy: a teenager's discovery, and an open question that remains open

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In plain words

Gauss was so struck by constructing the $17$ -gon that, according to his own later account, it was the discovery that convinced the 19-year-old to become a mathematician rather than a philologist; he asked to have a regular $17$ -gon engraved on his tombstone (the stonemason reportedly declined, fearing it would look too much like a circle).

Only five Fermat primes are known today — $3, 5, 17, 257, 65537$ — and mathematicians have checked many further candidates $F_5, F_6, \ldots$ without finding a sixth; whether infinitely many exist, or whether five is the complete list forever, is a genuinely open question in number theory.

F_0=3,\ F_1=5,\ F_2=17,\ F_3=257,\ F_4=65537

Detailed analysis

Gauss's discovery, recorded in his diary on 30 March 1796, was the first entry of what became a lifelong mathematical diary; he later wrote that this result — constructing a shape no one had linked to compass and straightedge in over two thousand years — settled his choice of career toward mathematics over philology. He asked for a regular $17$ -gon to be engraved on his gravestone; the mason judged it would be indistinguishable from a circle and used a $17$ -pointed star instead.

The five known Fermat primes are $F_0=3,\ F_1=5,\ F_2=17,\ F_3=257,\ F_4=65537$ (Fermat numbers $F_s=2^{2^s}+1$ ); Euler showed already in 1732 that the next candidate, $F_5=2^{32}+1=4294967297$ , is composite ( $=641\times6700417$ ), and every $F_s$ checked since, up through very large $s$ , has also been found composite. Whether any further Fermat primes exist is unknown — an open problem inherited directly from this proof.

Along with resolving angle trisection and doubling the cube in the very same paper, Wantzel's 1837 completion of Gauss's theorem meant three of antiquity's classical construction problems had been fully and rigorously settled by field-degree arguments developed just before Galois theory took its modern shape; only squaring the circle remained, awaiting Lindemann's 1882 proof that $\pi$ is transcendental.