Rational reciprocity law

In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity.

As an example, there are rational biquadratic and octic reciprocity laws. 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 4, such that (p|q)₂ = (q|p)₂ = +1. Let p = a² + b² and q = A² + B² with aA odd. Then

$(p|q)_4 (q|p)_4 = (-1)^{(q-1)/4} (aB-bA|q)_2 \ .$

If in addition p and q are congruent to 1 modulo 8, let p = c² + 2d² and q = C² + 2D². Then

$(p|q)_8 = (q|p)_8 = (aB-bA|q)_4 (cD-dC|q)_2 \ .$

References

• Burde, K. (1969), "Ein rationales biquadratisches Reziprozitätsgesetz", J. Reine Angew. Math. (in German) 235: 175–184, Zbl 0169.36902
• Lehmer, Emma (1978), "Rational reciprocity laws", The American Mathematical Monthly 85 (6): 467–472, doi:10.2307/2320065, ISSN 0002-9890, MR 0498352, Zbl 0383.10003
• Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp. 153–183, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
• 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, MR 0414467, Zbl 0311.10004