Editing Gaussian integer (section)

===Residue class fields===
The residue class ring modulo a Gaussian integer {{math|''z''<sub>0</sub>}} is a [[field (mathematics)|field]] if and only if <math>z_0</math> is a Gaussian prime.

If {{math|''z''<sub>0</sub>}} is a decomposed prime or the ramified prime {{math|1 + ''i''}} (that is, if its norm {{math|''N''(''z''<sub>0</sub>)}} is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, {{math|''N''(''z''<sub>0</sub>)}}). It is thus [[isomorphic]] to the field of the integers modulo {{math|''N''(''z''<sub>0</sub>)}}.

If, on the other hand, {{math|''z''<sub>0</sub>}} is an inert prime (that is, {{math|''N''(''z''<sub>0</sub>) {{=}} ''p''<sup>2</sup>}} is the square of a prime number, which is congruent to 3 modulo 4), then the residue class field has {{math|''p''<sup>2</sup>}} elements, and it is an [[field extension|extension]] of degree 2 (unique, up to an isomorphism) of the [[prime field]] with {{math|''p''}} elements (the integers modulo {{math|''p''}}).