Pattern for minimal polynomials of $\frac {\sin(\frac{(n-2) \pi}{n})}{\sin(\frac{\pi}{n})}$?

Question

I was studying distances between vertices of regular polygons with side length 1.

And thus I came across the numbers

$$\frac {\sin(\frac{(n-2) \pi}{n})}{\sin(\frac{\pi}{n})}=2\cos(\frac{\pi}{n})$$

Since they are the smallest distance between vertices in a regular n-polygon apart from the sides.

In a pentagon this gives us the golden mean.

Sometimes these numbers can thus be expressed by radicals.

But I think this fails for a prime $n \gt5$ ?

Make that last sentence question 1.

But the main question ( question 2 ) here is :

What are the minimum polynomials of $\frac {\sin(\frac{(n-2) \pi}{n})}{\sin(\frac{\pi}{n})}$ ? And is there a pattern in it ? Is there a recursion ? Are they named polynomials ? Is the degree simply $\lfloor\frac{n}{2}\rfloor$ ?

I am particularly interested in the case when $n$ is a prime .

Answer 1

The expression is, as noted by others, $2\cos \frac{\pi}{n}= 2 \cos\frac{2\pi}{2n}$. In general, the minimal polynomial of $2 \cos\frac{2\pi}{m}$ for $m \ge 3$ is obtained as follows:

Note that $2 \cos \frac{2 \pi}{m} = \zeta_m + \frac{1}{\zeta_m}$, where $\zeta_m = e^{\frac{2 \pi i}{m}}$. Now the minimal polynomial for $\zeta_m$ is the cyclotomic polynomial $\Phi_m(z)$. It has degree $\phi(m)$ ( $\phi$ the Euler's totient function), even, and is a palyndromic polynomial ( for $m\ge 3$). Therefore, we can write $$\Phi_m(z) = z^{\frac{\phi(m)}{2}} \cdot \Psi_m(z+\frac{1}{z})$$ where $\Psi_m$ is a polynomial of degree $\frac{\phi(m)}{2}$. One sees that $2 \cos \frac{2\pi}{m}$ is a root of $\Psi_m$. It turns out that $\Psi_m$ is irreducible, so $\Psi_m$ is the minimal polynomial of $2\cos \frac{2\pi}{m}$.

As an example, the minimal polynomial of $2\cos \frac{\pi}{17}=2\cos\frac{2 \pi}{34}$ is $$x^8 - x^7 - 7 x^6 + 6 x^5 + 15 x^4 - 10 x^3 - 10 x^2 + 4 x + 1 $$

Answer 2

Your numbers simplify to $2\cos(\frac{\pi}{n})$, and this article answers exactly the questions you're asking. But your conjecture is incorrect, as Fermat primes (and their products) allow radical solutions.

We have $\epsilon = \sqrt{17+\sqrt{17}}$, $\epsilon^* = \sqrt{17-\sqrt{17}}$, $\alpha = \sqrt{34 + 6\sqrt{17}+ \sqrt{2}(\sqrt{17}-1)\epsilon^* - 8\sqrt{2}\epsilon}$ and $\cos(\frac{\pi}{17}) = \frac{1}{8}\sqrt{2}\sqrt{15 + \sqrt{17} + \sqrt{2}(\alpha + \epsilon^*)}$, according to Mathworld.

There are strong geometric relations here, as it's related to the fact that the regular 17-gon is constructible.

Part of your second question is answered in this article. The minimal algebraic order of $\cos({\frac{\pi}{n}})$ is $\phi(n)$ where $\phi$ is the totient function.