Definition:Quadratic Residue/Character

Definition

Let $p$ be an odd prime.

Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$.

$a$ is either a quadratic residue or a quadratic non-residue of $p$.

Whether it is or not is known as the quadratic character of $a$ modulo $p$.

Also see

• Congruent Integers are of same Quadratic Character

Therefore it is sufficient to consider the quadratic character of the non-zero least positive residues modulo $p$.