Octic reciprocity explained

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( (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( (pq\right))/(_{4}) = \left( (qp\right))/(_{4}) = +1 . Let p = a² + b² = c² + 2d² and q = A² + B² = C² + 2D², with aA odd. Then\left( (pq\right))/(\left( _{8}) (qp\right))/(_{8}) = \left( (aB - bA)/(q\right)) _{4} \left( (cD - dC)/(q\right)) _{2} .

See also

• Artin reciprocity
• Eisenstein reciprocity