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 ${\displaystyle \left({\frac {x}{p}}\right)_{k}}$ 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 ${\displaystyle \left({\frac {p}{q}}\right)_{4}=\left({\frac {q}{p}}\right)_{4}=+1.}$ Let p = a² + b² = c² + 2d² and q = A² + B² = C² + 2D², with aA odd. Then

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