In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or âÂÂ1 rather than a general root of unity.
As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (x|p)<sub>k</sub> 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)<sub>2</sub> = (q|p)<sub>2</sub> = +1. Let p = a<sup>2</sup> + b<sup>2</sup> and q = A<sup>2</sup> + B<sup>2</sup> with aA odd. Then
If in addition p and q are congruent to 1 modulo 8, let p = c<sup>2</sup> + 2d<sup>2</sup> and q = C<sup>2</sup> + 2D<sup>2</sup>. Then