Editing Square root (section)

==In integral domains, including fields==
Each element of an [[integral domain]] has no more than 2 square roots. The [[difference of two squares]] identity {{math|1=''u''<sup>2</sup> − ''v''<sup>2</sup> = (''u'' − ''v'')(''u'' + ''v'')}} is proved using the [[commutative ring|commutativity of multiplication]]. If {{mvar|u}} and {{mvar|v}} are square roots of the same element, then {{math|1=''u''<sup>2</sup> − ''v''<sup>2</sup> = 0}}. Because there are no [[zero divisors]] this implies {{math|1=''u'' = ''v''}} or {{math|1=''u'' + ''v'' = 0}}, where the latter means that two roots are [[additive inverse]]s of each other. In other words if an element a square root {{mvar|u}} of an element {{mvar|a}} exists, then the only square roots of {{mvar|a}} are {{mvar|u}} and {{mvar|&minus;u}}. The only square root of 0 in an integral domain is 0 itself.

In a field of [[characteristic (algebra)|characteristic]]&nbsp;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 {{math|1=&minus;''u'' = ''u''}}. If the field is [[finite field|finite]] of characteristic&nbsp;2 then every element has a unique square root. In a [[field (mathematics)|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]] {{mvar|p}}, let {{math|1=''q'' = ''p''<sup>''e''</sup>}} for some positive integer {{mvar|e}}. A non-zero element of the field {{math|[[finite field|'''F'''<sub>''q''</sub>]]}} with {{mvar|q}} elements is a [[quadratic residue]] if it has a square root in {{math|'''F'''<sub>''q''</sub>}}. Otherwise, it is a quadratic non-residue. There are {{math|(''q'' − 1)/2}} quadratic residues and {{math|(''q'' − 1)/2}} quadratic non-residues; zero is not counted in either class. The quadratic residues form a [[group (mathematics)|group]] under multiplication. The properties of quadratic residues are widely used in [[number theory]].