Editing Gaussian integer (section)

==Unique factorization==
As for every [[unique factorization domain]], every Gaussian integer may be factored as a product of a [[unit (ring theory)|unit]] and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with a corresponding change of the unit factor).

If one chooses, once for all, a fixed Gaussian prime for each [[equivalence class]] of associated primes, and if one takes only these selected primes in the factorization, then one obtains a prime factorization which is unique up to the order of the factors. With the [[#selected associates|choices described above]], the resulting unique factorization has the form
:<math>u(1+i)^{e_0}{p_1}^{e_1}\cdots {p_k}^{e_k},</math>
where {{math|''u''}} is a unit (that is, {{math|''u'' ∈ {1, −1, ''i'', −''i''}{{void}}}}), {{math|''e''<sub>0</sub>}} and {{math|''k''}} are nonnegative integers, {{math|''e''<sub>1</sub>, …, ''e<sub>k</sub>''}} are positive integers, and {{math|''p''<sub>1</sub>, …, ''p<sub>k</sub>''}} are distinct Gaussian primes such that, depending on the choice of selected associates,
*either {{math|''p<sub>k</sub>'' {{=}} ''a<sub>k</sub>'' + ''ib<sub>k</sub>''}} with {{math|''a''}} odd and positive, and {{math|''b''}} even,
*or the remainder of the Euclidean division of {{math|''p<sub>k</sub>''}} by {{math|2 + 2''i''}} equals 1 (this is Gauss's original choice<ref>{{harvtxt|Gauss|1831|p=546}}</ref>).
An advantage of the second choice is that the selected associates behave well under products for Gaussian integers of odd norm. On the other hand, the selected associate for the real Gaussian primes are negative integers. For example, the factorization of 231 in the integers, and with the first choice of associates is {{nowrap|3 × 7 × 11}}, while it is {{nowrap|(−1) × (−3) × (−7) × (−11)}} with the second choice.