Eisenstein integer

English

Etymology

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

Noun

Eisenstein integer (plural Eisenstein integers)

1. (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.

Hypernyms

• quadratic integer
  • algebraic integer

Hyponyms

• Eisenstein prime

See also

• Gaussian integer