= Regular heptagon
{title2=$n=7$}
A regular heptagon is a <regular polygon> with seven sides. Its unit-circle vertices have complex coordinates $e^{2\pi ik/7}$ for $k=0,\ldots,6$. It cannot be constructed from a unit segment by a <straightedge and compass construction>: the first coordinate $\cos(2\pi/7)$ has degree three over $\mathbb Q$, since the seventh <cyclotomic field> has degree six and degree two over its real subfield. Every finite constructible coordinate extension has power-of-two degree, which cannot contain a degree-three subextension by the <tower law>.
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