Definition:Quadratic Residue/Warning

Quadratic Residue: Warning

The definition of the quadratic residue must be qualified as follows:

The case where $a = 0$ has been excluded from the definition, despite the fact that $0 = 0^2$ and so is definitely a square.

The case where $p = 2$ is also excluded, where the only non-zero residue $1$ is also a square.

The main reason for this is so that some useful results can be expressed in a convenient form.

For example, this means that from Number of Quadratic Residues of Prime the number of quadratic residues of $p$ is always equal to $\dfrac {p - 1} 2$, which is the same as the number of quadratic non-residues.