Editing Gaussian integer (section)

==Principal ideals==
Since the ring {{math|''G''}} of Gaussian integers is a Euclidean domain, {{math|''G''}} is a [[principal ideal domain]], which means that every [[ideal (ring theory)|ideal]] of {{mvar|G}} is [[principal ideal|principal]]. Explicitly, an [[ideal (ring theory)|ideal]] {{mvar|I}} is a subset of a ring {{mvar|R}} such that every sum of elements of {{mvar|I}} and every product of an element of {{mvar|I}} by an element of {{mvar|R}} belong to {{mvar|I}}. An ideal is [[principal ideal|principal]] if it consists of all multiples of a single element {{math|''g''}}, that is, it has the form
:<math>\{gx\mid x\in G\}.</math>
In this case, one says that the ideal is ''generated'' by {{math|''g''}} or that {{math|''g''}} is a ''generator'' of the ideal.

Every ideal {{math|''I''}} in the ring of the Gaussian integers is principal, because, if one chooses in {{math|''I''}} a nonzero element {{math|''g''}} of minimal norm, for every element {{math|''x''}} of {{math|''I''}}, the remainder of Euclidean division of {{math|''x''}} by {{math|''g''}} belongs also to {{math|''I''}} and has a norm that is smaller than that of {{math|''g''}}; because of the choice of {{math|''g''}}, this norm is zero, and thus the remainder is also zero. That is, one has {{math|1=''x'' = ''qg''}}, where {{math|''q''}} is the quotient.

For any {{math|''g''}}, the ideal generated by {{math|''g''}} is also generated by any ''associate'' of {{math|''g''}}, that is, {{math|''g'', ''gi'', −''g'', −''gi''}}; no other element generates the same ideal. As all the generators of an ideal have the same norm, the ''norm of an ideal'' is the norm of any of its generators.

{{Anchor|selected associates}}In some circumstances, it is useful to choose, once for all, a generator for each ideal. There are two classical ways for doing that, both considering first the ideals of odd norm. If the {{math|1=''g'' = ''a'' + ''bi''}} has an odd norm {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}}, then one of {{math|''a''}} and {{math|''b''}} is odd, and the other is even. Thus {{math|''g''}} has exactly one associate with a real part {{math|''a''}} that is odd and positive. In his original paper, [[Gauss]] made another choice, by choosing the unique associate such that the remainder of its division by {{math|2 + 2''i''}} is one. In fact, as {{math|1=''N''(2 + 2''i'') = 8}}, the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit. Multiplying {{math|''g''}} by the inverse of this unit, one finds an associate that has one as a remainder, when divided by {{math|2 + 2''i''}}.

If the norm of {{math|''g''}} is even, then either {{math|1=''g'' = 2<sup>''k''</sup>''h''}} or {{math|1=''g'' = 2<sup>''k''</sup>''h''(1 + ''i'')}}, where {{math|''k''}} is a positive integer, and {{math|''N''(''h'')}} is odd. Thus, one chooses the associate of {{math|''g''}} for getting a {{math|''h''}} which fits the choice of the associates for elements of odd norm.