quadratic reciprocity rule

Theorem (Law of Quadratic Reciprocity).

Let $p$ and $q$ be two distinct odd primes. Then:

$$\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)={(-1)}^{(p-1)(q-1)/4}$$

where $\mathrm{\left(}\frac{\mathrm{\cdot }}{\mathrm{\cdot }}\mathrm{\right)}$ is the Jacobi (http://planetmath.org/JacobiSymbol) symbol (or Legendre symbol).

The following is an equivalent formulation of the Law of Quadratic Reciprocity:

Theorem (Quadratic Reciprocity (second form)).

Let $p\mathrm{,}q$ be distinct odd primes. Then:

1. $\left({\displaystyle \frac{p}{q}}\right)=\left({\displaystyle \frac{q}{p}}\right)$ if one of $p$ or $q$ is congruent to $1$ modulo $4$;

2. $\left({\displaystyle \frac{p}{q}}\right)=-\left({\displaystyle \frac{q}{p}}\right)$ if both $p$ and $q$ are congruent to $3$ modulo $4$.