Communications of the
Korean Mathematical Society CKMS

In this article we show a recursive method to compute the coefficients of the minimal polynomial of $\cos (2\pi / n) $ explicitly for $n\geq 3$. The recursion is not on $n$ but on the coefficient index. Namely, for a given $n,$ we show how to compute $e_{i}$ of the minimal polynomial $ \sum_{i=0}^{d}(-1)^ie_{i}x^{d-i}$ for $i\geq 2$ with initial data $e_{0}=1,e_{1}=\mu(n)/2,$ where $\mu(n)$ is the M\"{o}bius function.

Keywords: minimal polynomial, cyclotomic polynomial, algebraic number, Ramanujan sum, cosine

MSC numbers: 11B83