Quintic reciprocity

Helou, Charles

doi:10.1090/S0002-9939-1993-1155597-X

Quintic reciprocity
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Proc. Amer. Math. Soc. 117 (1993), 877-884 Request permission

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

Beweis der allgemeinsten Reciprocitatsgesetze zwischen reellen und complexen Zahlen

Charles Helou, On rational reciprocity, Proc. Amer. Math. Soc. 108 (1990), no. 4, 861–866. MR 1007498, DOI 10.1090/S0002-9939-1990-1007498-2
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
Emma Lehmer, The quintic character of $2$ and $3$, Duke Math. J. 18 (1951), 11–18. MR 40338
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Kenneth S. Williams, Explicit criteria for quintic residuacity, Math. Comp. 30 (1976), no. 136, 847–853. MR 412089, DOI 10.1090/S0025-5718-1976-0412089-9