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Sent: Sunday, July 31, 2016 at 4:40 PM From: apgoucher@gmx.com To: "James Propp" <jamespropp@gmail.com> Subject: Re: [math-fun] Reentrant polygons with area zero

Consider the space V of rational linear combinations of pth roots of unity. This space has dimension p - 1, since the degree of the minimal poly of omega is p - 1.

It's also the direct sum of the spaces Re(V) and Im(V). Each of these has dimension at most (p - 1)/2, and equality must hold since dim(V) = dim(Re(V)) + dim(Im(V)).

The result follows from dim(Im(V)) = (p - 1)/2.

Best wishes,

Adam P. Goucher

-----Original message----- Sent: Sunday, 31 July 2016 at 13:30:31 From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Reentrant polygons with area zero It's a nice approach, but I'm missing a step near the end. Why is it true that

When n is prime, the *only* integer linear relationships between

...

the imaginary parts of roots of unity are generated by relationships of the form:

Im[w^k] + Im[w^-k] = 0

?

Note that Adam is talking about the *nontrivial* roots of unity. After all, the relation Im[w^0] = 0 has an odd number of terms. But this doesn't mar Adam's argument because w^(a_0 - a_1), ... are all unequal to 1 (which is I suppose what Adam was getting at by saying "Note that none of these terms are zero").

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