The quadratic reciprocity law can be formulated in terms of the Hilbert symbol (a,b)_v where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p-adic absolute values for primes p). The Hilbert symbol (a,b)_v is 1 or −1. It is defined to be 1 if and only if the equation ax^2+by^2=z^2 has a solution in the completion of the rationals at v other than x=y=z=0. The Hilbert reciprocity law states that (a,b)_v, for fixed a and b and varying v, is 1 for all but finitely many v and the product of (a,b)_v over all v is 1. (This formally resembles the residue theorem from complex analysis.)
