Exploring cyclotomic polynomials

Extract

Cyclotomic polynomials, (x^p - l)/(x-1), have (p-1) zeros exp(2πiq/p) (q = 1, 2, ... ,p - 1). We consider related polynomials whose zeros are given by cos(qπ/p) (q = 1, 2, ... ,p - 1) and indicate their relation to polynomials P_n (ζ) that satisfy the ‘double angle’ condition P_n(2ζ² - l) = P_n(ζ)P_n(-ζ). The double angle condition is generalised to a chain of double angle conditions Q_i(2ζ² - 1) = Q_i+1(ζ)Q_i+1(-ζ), i = 1, 2, ...,n(Q₁ ≡ Q_n+1) For purposes of illustration, we will focus on P₄(ζ), demonstrating how construction of the fourth order polynomial subject to the double angle condition yields an equation for cos . We also discuss more general features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple geometrical construction can be used to derive a polynomial for cos .