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Exploring cyclotomic polynomials

Published online by Cambridge University Press:  01 August 2016

D. G. C. Mckeon  and
T. N. Sherry
D. G. C. Mckeon
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada. e-mail: DGMCKE02@JULIAN.UWO.CA
T. N. Sherry
Affiliation:
Department of Mathematical Physics, National University of Ireland, Galway, Ireland. e-mail: TOM.SHERRY@NUIGALWAY.IE
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Extract

Cyclotomic polynomials, (xp - 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 Pn (ζ) that satisfy the ‘double angle’ condition Pn(2ζ2 - l) = Pn(ζ)Pn(-ζ). The double angle condition is generalised to a chain of double angle conditions Qi(2ζ2 - 1) = Qi+1(ζ)Qi+1(-ζ), i = 1, 2, ...,n(Q1 ≡ Qn+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 .

Type
Articles
Information
The Mathematical Gazette , Volume 85 , Issue 502 , March 2001 , pp. 59 - 65
Copyright
Copyright © The Mathematical Association 2001

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References

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