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
If in addition p and q are congruent to 1 modulo 8, let p = c2 + 2d2 and q = C2 + 2D2. Then
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, JSTOR 2320065, 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
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