On the constant term of the minimal polynomial of $\cos \left(\frac{2\pi}{n}\right)$ over Q

Abstract

The algebraic numbers $\cos \left(2\pi/n\right)$ and $2\cos \left(\pi/n\right)$ play an important role in the theory of discrete groups, especially in modular group and Hecke groups, and has many applications because of their relation with Chebycheff polynomials. There are some partial results in literature for the minimal polynomial of the latter number over rationals until 2012 when a complete solution was given in $\left[5\right]$. In this paper we determine theconstant term of the minimal polynomial of $\cos (\frac{2\pi}{n})$ over $\mathbb{Q}$ which is important in determining the congruence subgroups of Hecke groups, by a new method.