Octic reciprocity

Short description: Reciprocity law relating the residues of 8th powers modulo primes

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

[math]\displaystyle{ \left(\frac pq\right)_8 \left(\frac qp\right)_8 = \left(\frac{aB-bA}q\right)_4 \left(\frac{cD-dC}q\right)_2 \ . }[/math]

See also

• Artin reciprocity
• Eisenstein reciprocity

References

• Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Springer-Verlag, Berlin, pp. 289–316, ISBN 3-540-66957-4, https://books.google.com/books?id=EwjpPeK6GpEC 
• Williams, Kenneth S. (1976), "A rational octic reciprocity law", Pacific Journal of Mathematics 63 (2): 563–570, doi:10.2140/pjm.1976.63.563, ISSN 0030-8730, http://projecteuclid.org/euclid.pjm/1102867415