Cyclotomic resultants
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- by D. H. Lehmer and Emma Lehmer PDF
- Math. Comp. 48 (1987), 211-216 Request permission
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
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Paul Bachmann, Die Lehre von der Kreistheilung, B. G. Teubner, Leipzig, 1872, pp. 210-213, 224-230.
- Ronald J. Evans, The octic periodic polynomial, Proc. Amer. Math. Soc. 87 (1983), no. 3, 389–393. MR 684624, DOI 10.1090/S0002-9939-1983-0684624-2 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. 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.
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 211-216
- MSC: Primary 11T21
- DOI: https://doi.org/10.1090/S0025-5718-1987-0866110-1
- MathSciNet review: 866110