I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial
\begin{equation}
x^{8}-7x^{6}+14x^{4}-8x^{2}+1
\end{equation}
does not have any root of the form
\begin{equation}
2\cos\frac{(2k+1)\pi}{2n} \quad k\in \lbrace 0,1,\cdots , n-1 \rbrace , n \in \mathbb{E}.
\end{equation}
Can anybody help me?


Bests.



How to prove the following polynomial does not have root of a special form?
The degree of the minimal polynomial of your cos is bounded below by phi(n)/2 since it generates a real field inside the cyclotomic field. If phi(n)/2 is bigger than 8 you are done. For small n you can check one by one with computer (or write the minimal polynomial and see it doesn’t divide that polynomial or compute discriminant, which is an easy thing for cosine as the real part of a root of unity)


