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 *e*th 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**, 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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DOI:
https://doi.org/10.1090/S0025-5718-1987-0866110-1

Article copyright:
© Copyright 1987
American Mathematical Society