Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Quintic reciprocity


Author: Charles Helou
Journal: Proc. Amer. Math. Soc. 117 (1993), 877-884
MSC: Primary 11A15; Secondary 11L99
DOI: https://doi.org/10.1090/S0002-9939-1993-1155597-X
MathSciNet review: 1155597
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An expression for the rational inversion factor of the power residue symbol, of odd prime exponent $ n \equiv 1\;(\bmod \,4)$ (mod 4), is given. It is applied to the quintic case, where the resulting expression involves only a rational quadratic form representation of primes and the power residue character of Jacobi sums. A reciprocity relation for Jacobi sums is then deduced, for $ n = 5$, and conjectured to hold for all odd prime exponents $ n$.


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

  • [1] G. Eisenstein, Beweis der allgemeinsten Reciprocitatsgesetze zwischen reellen und complexen Zahlen, Math. Werke, 2, Chelsea, New York, 1975, pp. 712-721.
  • [2] C Helou, On rational reciprocity, Proc. Amer. Math. Soc. 108 (1990), 861-866. MR 1007498 (90g:11007)
  • [3] K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., Springer, New York, 1990. MR 1070716 (92e:11001)
  • [4] E. Lehmer, The quintic character of $ 2$ and $ 3$, Duke Math. J. 18 (1951), 11-18. MR 0040338 (12:677a)
  • [5] L. Washington, Introduction to cyclotomic fields, Springer, New York, 1982. MR 718674 (85g:11001)
  • [6] K. S. Williams, Explicit criteria for quintic residuacity, Math. Comp. 30 (1976), 847-853. MR 0412089 (54:218)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11A15, 11L99

Retrieve articles in all journals with MSC: 11A15, 11L99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1155597-X
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society