Octic reciprocity

In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity.

There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol (x|p)_k to be +1 if x is a k-th power modulo the prime p and -1 otherwise. Let p and q be distinct primes congruent to 1 modulo 8, such that (p|q) = (q|p) = +1. Let p = a² + b² = c² + 2d² and q = A² + B² = C² + 2D², with aA odd. Then

${\displaystyle (p|q)_{8}=(q|p)_{8}=(aB-bA|q)_{4}(cD-dC|q)_{2}\ .}$