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Mathematics of Computation

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Cyclotomic resultants

Authors: D. H. Lehmer and Emma Lehmer
Journal: Math. Comp. 48 (1987), 211-216
MSC: Primary 11T21
MathSciNet review: 866110
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Abstract: This paper examines the eth power character of the divisors of two cyclotomic period polynomials of degree $ {e_1}$ and $ {e_2}$. The special cases $ {e_1} = 2$ and $ {e_2} = 3,4$, are considered in detail. As corollaries one finds new conditions for cubic and quartic residuacity.

The computational method consists in representing cyclotomic numbers in the form $ {c_1}\zeta + {c_2}{\zeta ^2} + \cdots + {c_{p - 1}}{\zeta ^{p - 1}}$, where $ \zeta = {e^{2\pi i/p}}$. Multiplication is reduced to addition and subtraction, which are carried out in a multi-precision system.

References [Enhancements On Off] (What's this?)

  • [1] Paul Bachmann, Die Lehre von der Kreistheilung, B. G. Teubner, Leipzig, 1872, pp. 210-213, 224-230.
  • [2] Ronald J. Evans, "The octic period polynomial," Proc. Amer. Math. Soc., v. 87, 1983, pp. 389-393. MR 684624 (84b:10055)
  • [3] E. E. Kummer, "Über die Divisoren gewisser Formen der Zahlen welche aus der Theorie der Kreistheilung entstehen," J. Reine Angew. Math., v. 30, 1846, pp. 107-116, Collected papers, v. 1, pp. 193-239.
  • [4] J. J. Sylvester, "On the multisection of roots of unity," Johns Hopkins Univ. Circular, v. 1, 1881, pp. 150-151, Collected papers, v. 3, pp. 477-478.

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Article copyright: © Copyright 1987 American Mathematical Society

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