Editing Gaussian integer (section)

===Examples===

*There are exactly two residue classes for the modulus {{math|1 + ''i''}}, namely {{math|{{overline|0}} {{=}} {0, ±2, ±4,…,±1 ± ''i'', ±3 ± ''i'',…}{{void}}}} (all multiples of {{math|1 + ''i''}}), and {{math|{{overline|1}} {{=}} {±1, ±3, ±5,…, ±''i'', ±2 ± ''i'',…}{{void}}}}, which form a checkerboard pattern in the complex plane. These two classes form thus a ring with two elements, which is, in fact, a [[field (mathematics)|field]], the unique (up to an isomorphism) field with two elements, and may thus be identified with the [[modular arithmetic|integers modulo 2]]. These two classes may be considered as a generalization of the partition of integers into even and odd integers. Thus one may speak of ''even'' and ''odd'' Gaussian integers (Gauss divided further even Gaussian integers into ''even'', that is divisible by 2, and ''half-even'').
*For the modulus 2 there are four residue classes, namely {{math|{{overline|0}}, {{overline|1}}, {{overline|''i''}}, {{overline|1 + ''i''}}}}. These form a ring with four elements, in which {{math|1=''x'' = −''x''}} for every {{math|''x''}}. Thus this ring is not [[isomorphic]] with the ring of integers modulo 4, another ring with four elements. One has {{math|{{overline|1 + ''i''}}<sup>2</sup> {{=}} {{overline|0}}}}, and thus this ring is not the [[finite field]] with four elements, nor the [[direct product]] of two copies of the ring of integers modulo 2.
*For the modulus {{math|2 + 2i {{=}} (''i'' − 1)<sup>3</sup>}} there are eight residue classes, namely {{math|{{overline|0}}, {{overline|±1}}, {{overline|±''i''}}, {{overline|1 ± ''i''}}, {{overline|2}}}}, whereof four contain only even Gaussian integers and four contain only odd Gaussian integers.