Rational residuacity of primes

Abstract

The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of ‫(ޑ‬ζ p ). As an example, a new generalization of Burde's law is provided.

Computational aspects of rational residuosity

“…Proof. One easily observes that the pairs (3, 1) and (1,3) are the only solutions to the congruence N ≡ XY mod 4. Therefore, one of the prime factors of N is congruent to 3 modulo 4, whereas the other one is congruent to 1 modulo 4.…”

“…Burde's and Scholz's reciprocity laws for the biquadratic case and their generalizations for the octic and for higher power cases are well-known. We refer the reader to [5], to [1,Theorem 3] and to [3,Theorem 3.1]. For a general overview on power residue symbols in number fields and other rational power residue symbols, we suggest the survey [6].…”

“…But in general, this is not the case. Therefore, we are interested in methods to compute this symbol and solve the problem without knowing the factorization of n. We observe that there is a polynomial-time algorithm to solve CRS (1), since the problem reduces to the computation of Jacobi's symbol, which can be done efficiently by using the quadratic reciprocity law. For k > 1, the problem appears to be open and the currently known results and reciprocity laws do not apply.…”