Eisenstein integer

English

Named after Gotthold Eisenstein (1823–1852), German mathematician.

Eisenstein integer (plural Eisenstein integers)

(algebra) A complex number of the form $a + b ω$ , where a and b are integers and ω is defined by the following two rules: (1) $ω^{3} = 1$ and (2) $1 + ω + ω^{2} = 0$ ; an element of the Euclidean domain $ℤ [ω]$ .
To divide an Eisenstein integer $a + b ω$ by another Eisenstein integer $c + d ω$ , notice that $(c + d ω) (c + d) (c + d ω^{2}) = c^{3} + d^{3}$ ; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by $(c + d) (c + d ω^{2})$ , then simplify.