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 $$ \left(\frac xp\right)_k $$ 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 $$ \left(\frac pq\right)_4 = \left(\frac qp\right)_4 = +1 .$$ Let p = a2 + b2 = c2 + 2d2 and q = A2 + B2 = C2 + 2D2, with aA odd. Then


 * $$ \left(\frac pq\right)_8 \left(\frac qp\right)_8 = \left(\frac{aB-bA}q\right)_4 \left(\frac{cD-dC}q\right)_2 \ .$$