Here is the problem:
Let P sub 0, ....., P sub n-1 be n equally spaced points on the unit circle. Compute the product of the distances from P sub 0 (naught) to all the remaining points.

I do not understand. What distances and what products.

First we know that,
P_k = exp((2pi/n)*ki)

Now what?

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This is my understanding.

In that case the product is equal to "n" itself.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

Correct! What is the proof?

Not an easy question to answer.
But we can begin by simplifing.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

I am going in unit circles (get the pun) with this problem.

But I managed to make this problem more complicated than it should be. Instead of working with cyclotonomic polynomials, as I should have. I reduced the problem to trigonometry which makes it more difficult.


I am sure you know what I am about to say but let me just post it.

A cyclotonomic polynomial is,
\Phi_n(z) = 1+z+z^2+...+z^n

We can factor them as,
(z-z_1)(z-z_2)+...+(z-_n)
Where,
z_1,z_2,...,z_n
Are points on the regular (n+1)-gon inscribed in the unit circle in the complex plane (except unity).

Basically, I am saying this is the cyclotonomic polynomial of degree n evaluated at 1.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran