Unless I am misunderstanding, so is 1/17.
ETA: Oh, the irony!
Definitively Answered…Flawless Victory.
Maybe I misunderstood something here. 1/7 is indeed constructible. Any rational division of a straight line is.
A 7-gon certainly is not constructible. And that is what I thought was being asked.
No. The 17-gon is constructible, but 1/17 is cyclical. Same for 257 and 65537. In fact, whenever the n-gon is constructible for a prime n > 5, we have that 1/n is cyclical.
(By “the n-gon”, I mean “the regular n-gon” throughout, of course).
Also, a proof sketch (though perhaps there’s something much simpler?): Using the existence of primitive roots modulo any prime and the observation that every odd multiple of a generator of a cyclic group of even order also generates the group, we find that x is a primitive root modulo a Fermat prime just in case x is not a quadratic residue modulo that Fermat prime. Sufficiently large Fermat primes must be of the form 16^n + 1, and thus have last decimal digit 7, and thus, by quadratic reciprocity, 10 will not be a quadratic residue of those primes. Thus, Fermat primes have cyclical reciprocals. Finally, invoke Gauss for the connection between Fermat primes and constructible regular polygons.
[This depends on the fact that 10 = ten, mind you. This is not base-independent]
Sorry, I meant “of a finite cyclic group of order a power of two”.