Editing Gaussian integer (section)

==Historical background==
The ring of Gaussian integers was introduced by [[Carl Friedrich Gauss]] in his second monograph on [[quartic reciprocity]] (1832).<ref>{{harvtxt|Kleiner|1998}}</ref> The theorem of [[quadratic reciprocity]] (which he had first succeeded in proving in 1796) relates the solvability of the congruence {{math|''x''<sup>2</sup> ≡ ''q'' (mod ''p'')}} to that of {{math|''x''<sup>2</sup> ≡ ''p'' (mod ''q'')}}. Similarly, cubic reciprocity relates the solvability of {{math|''x''<sup>3</sup> ≡ ''q'' (mod ''p'')}} to that of {{math|''x''<sup>3</sup> ≡ ''p'' (mod ''q'')}}, and biquadratic (or quartic) reciprocity is a relation between {{math|''x''<sup>4</sup> ≡ ''q'' (mod ''p'')}} and {{math|''x''<sup>4</sup> ≡ ''p'' (mod ''q'')}}. Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).

In a footnote he notes that the [[Eisenstein integer]]s are the natural domain for stating and proving results on [[cubic reciprocity]] and indicates that similar extensions of the integers are the appropriate domains for studying higher reciprocity laws.

This paper not only introduced the Gaussian integers and proved they are a unique factorization domain, it also introduced the terms norm, unit, primary, and associate, which are now standard in algebraic number theory.