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Definition 2024

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