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In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable.

Shouldn't this be $\frac{\pi}{n}$ instead of $\frac{2\pi}{n}$ ? (since each angle of a regular polygon with $n$ sides is $\frac{\pi}{n}$ and not $\frac{2\pi}{n}$)

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It doesn't matter, an angle $\theta$ is constructible if and only if $2 \theta$ is. –  Will Jagy Sep 9 '12 at 22:30
    
What angle is in the regular $n$-gon is it you claim to be $\pi/n$? The only thing I can match it to is the angle between two neighboring diagonals at one of the corners. (Have you perhaps fooled yourself into thinking that the angle sum of every polygon is $\pi$? It is really $(n-2)\pi$). –  Henning Makholm Sep 9 '12 at 22:52

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As Will Jagy noted, those two are equivalent (in terms of constructibility), since doubling an angle and halving an angle are both legal constructions.

Another way to see that it is correct is by looking at the angles that appear on the circumscribed circle, which (at least to me...) seem to be more naturally related to construction of a regular polygon than the angles between the sides.

Even more so in algebraic context, where the vertex set of a regular $n$-gon is just the set of $n$-th roots of unity...

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