A NEW CHARACTERISATION FOR QUARTIC RESIDUACITY OF

Abstract

AbstractLet p be a prime with $p\equiv 1\pmod {4}$ . Gauss first proved that $2$ is a quartic residue modulo p if and only if $p=x^2+64y^2$ for some $x,y\in \Bbb Z$ and various expressions for the quartic residue symbol $(\frac {2}{p})_4$ are known. We give a new characterisation via a permutation, the sign of which is determined by $(\frac {2}{p})_4$ . The permutation is induced by the rule $x \mapsto y-x$ on the $(p-1)/4$ solutions $(x,y)$ to $x^2+y^2\equiv 0 \pmod {p}$ satisfying $1\leq x < y \leq (p-1)/2$ .