is positive definite. In integral domains, including fields Each element of an integral domain has no more than 2 square roots. The difference of two squares identity is proved using the commutativity of multiplication. If and are square roots of the same element, then . Because there are no zero divisors this implies or , where the latter means that two roots are additive inverses of each other. In other words if an element a square root of an element exists, then the only square roots of are and . The only square root of 0 in an integral domain is 0 itself. In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that . If the field is finite of characteristic 2 then every element has a unique square root. In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any. Given an odd prime number , let for some positive integer . A non-zero element of the field with elements is a quadratic residue if it has a square root in . Otherwise, it is a quadratic non-residue. There are quadratic residues and quadratic non-residues; zero is not counted in either class. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory. In rings in general Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3. Another example is provided by the ring of quaternions which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including , , and . In fact, the set of square roots of −1 is exactly A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in any multiple of is a square root of 0. Geometric construction of the square root The square root of a positive number