Cyclotomic resultants
Authors:
D. H. Lehmer and Emma Lehmer
Journal:
Math. Comp. 48 (1987), 211-216
MSC:
Primary 11T21
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866110-1
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 and
. The special cases
and
, 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 , where
. Multiplication is reduced to addition and subtraction, which are carried out in a multi-precision system.
- [1] Paul Bachmann, Die Lehre von der Kreistheilung, B. G. Teubner, Leipzig, 1872, pp. 210-213, 224-230.
- [2] Ronald J. Evans, The octic periodic polynomial, Proc. Amer. Math. Soc. 87 (1983), no. 3, 389–393. MR 684624, https://doi.org/10.1090/S0002-9939-1983-0684624-2
- [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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Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866110-1
Article copyright:
© Copyright 1987
American Mathematical Society