Historical Background
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity (1832) (see ). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence x2 ≡ q (mod p) to that of x2 ≡ p (mod q). Similarly, cubic reciprocity relates the solvability of x3 ≡ q (mod p) to that of x3 ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation between x4 ≡ q (mod p) and x4 ≡ 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 integers 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.
Read more about this topic: Gaussian Integer
Famous quotes containing the words historical and/or background:
“This seems a long while ago, and yet it happened since Milton wrote his Paradise Lost. But its antiquity is not the less great for that, for we do not regulate our historical time by the English standard, nor did the English by the Roman, nor the Roman by the Greek.... From this September afternoon, and from between these now cultivated shores, those times seemed more remote than the dark ages.”
—Henry David Thoreau (1817–1862)
“Silence is the universal refuge, the sequel to all dull discourses and all foolish acts, a balm to our every chagrin, as welcome after satiety as after disappointment; that background which the painter may not daub, be he master or bungler, and which, however awkward a figure we may have made in the foreground, remains ever our inviolable asylum, where no indignity can assail, no personality can disturb us.”
—Henry David Thoreau (1817–1862)