Quadratic Residue

Definition

Given \(a, n \in \mathbb{Z}\), \(a\) is a quadratic residue modulo \(n\) if (by definition):

\[ x^2 \equiv a \pmod n.\]

has a solution \(x \in \mathbb{Z}_p\).

Otherwise \(a\) is called a quadratic non-residue.

This is simply a generalisation of the notion of a square number to the ring \(\mathbb{Z}_n\).

Note that in the context of MATH3431, \(0\) is not considered to be either a quadratic residue or non-residue, and for the sake of the Legendre symbol and Euler's criterion it can be useful to exclude \(0\), so one must take care in the definition. Generally in these notes, definitions will be written to account for this, and \(0\) is a quadratic residue.