The constant term of the minimal polynomial of cos ( 2 π / n ) over ℚ

Musa Demirci,Ismail Naci Cangül

doi:10.1186/1687-1812-2013-77

Abstract

Let H(λq) be the Hecke group associated to λq =2 cos π q for q ≥ 3 integer. In this paper, we determine the constant term of the minimal polynomial of λq denoted by P ∗(x). MSC: 12E05; 20H05

Highlights

The Hecke groups H(λ) are defined to be the maximal discrete subgroups of PSL(, R) generated by two linear fractional transformationsT(z) = – and S(z) =, z z+λ where λ is a fixed positive real number
It is well known that H(λq) has a presentation as follows: H(λq) = T, S | T = Sq = I
In [ ], Cangul studied the minimal polynomials of the real part of ζ, i.e., of cos( π/n) over the rationals

Summary

Introduction

The Hecke groups H(λ) are defined to be the maximal discrete subgroups of PSL( , R) generated by two linear fractional transformations. In [ ], Cangul studied the minimal polynomials of the real part of ζ , i.e., of cos( π/n) over the rationals. He used a paper of Watkins and Zeitlin [ ] to produce further results. It is known that for n ∈ N ∪ { }, the nth Chebycheff polynomial, denoted by Tn(x), is defined by. For n ∈ N, Cangul denoted the minimal polynomial of cos( π/n) over Q by n(x). He obtained the following formula for the minimal polynomial n(x).

If p

Notice that there is a relation

Also note that π