Supriya Ghosh (Editor)

Rational reciprocity law

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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 ( a B − b A | 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 = ( a B − b A | q ) 4 ( c D − d C | q ) 2   .

References

Rational reciprocity law Wikipedia
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