Reciprocity and Jacobi sums

Abstract

Recently N. C. Ankeny derived a law of rth power reciprocity, where r is an odd prime: q is an rth power residue, modulo p = 1 (mod r), if and only if the rth power of the Gaussian sum (or Lagrange resolvent) r(), which depends upon p and r, is an rth power in GF(q f ), where q belongs to the exponent / (mod r). () r can be written as the product of algebraic integers known as Jacobi sums. Conditions in which the reciprocity criterion can be expressed in terms of a single Jacobi sum are presented in this paper.