Rational reciprocity law

Template:Short description 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

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation

Template:Numtheory-stub