Some remarks about the minimal polynomials of $4{\sin ^2}\left( {\frac{\pi }{n}} \right)$

Abstract

We show that the values of the minimal polynomials ${\phi_n}(x)$ of $4{\sin ^2}\left( {\frac{\pi }{n}} \right)$ for $x \in \left\{ {0,1,2,3,4} \right\}$ are intimately related to the prime factorization of $n.$