Legendre symbol

Definition

Suppose $p$ is a prime number and $a$ is an integer. The Legendre symbol of $a$ modulo $p$ , denoted $(\frac{a}{p})$ is defined to be:

$0$ if $p$ divides $a$ .
$1$ if $a$ is a quadratic residue modulo $p$ , i.e., $p$ does not divide $a$ and $x^{2} \equiv a (\mod p)$ has an integer solution for $x$ .
$- 1$ if $a$ is a quadratic nonresidue modulo $p$ , i.e., $p$ does not divide $a$ and $x^{2} \equiv a (\mod p)$ has no integer solution for $x$ .

The Legendre symbol makes sense only relative to a prime number, but it has a generalization called the Jacobi symbol which allows the modulus to be any integer. Thus, the Legendre symbol can also be defined as the restriction of the Jacobi symbol to cases where the modulus is prime.

The Legendre symbol is also a special case of a Dirichlet character modulo $p$ .

Facts

The Legendre symbol is multiplicative in its top portion if the prime is fixed. In other words:

$(\frac{a_{1} a_{2}}{p}) = (\frac{a_{1}}{p}) (\frac{a_{2}}{p})$

Quadratic reciprocity relates the Legendre symbol values of two distinct odd primes modulo each other.