TOPICS
Search

Quadratic Reciprocity

Quadratic reciprocity is the relation between the solvability of the congruences x^2=p (modq) and x^2=q (modp) for distinct odd primes p and q. In terms of Legendre symbols, the relation is

(p/q)(q/p)=(-1)^((p-1)(q-1)/4).

Thus p is a quadratic residue modulo q exactly when q is a quadratic residue modulo p, except when both primes are congruent to 3 modulo 4. The quadratic reciprocity theorem proves this relation.

See also

Legendre Symbol, Quadratic Reciprocity Theorem, Quadratic Residue, Reciprocity Theorem

Explore with Wolfram|Alpha

References

Ireland, K. and Rosen, M. "Quadratic Reciprocity." Ch. 5 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 50-65, 1990.

Cite this as:

Weisstein, Eric W. "Quadratic Reciprocity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuadraticReciprocity.html

Subject classifications