Eisenstein integer

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

Eisenstein integer (plural Eisenstein integers)

1. (algebra) A complex number of the form ${\displaystyle a+b\omega }$ , where a and b are integers and ω is defined by the following two rules: (1) ${\displaystyle \omega ^{3}=1}$ and (2) ${\displaystyle 1+\omega +\omega ^{2}=0}$ ; an element of the Euclidean domain ${\displaystyle \mathbb {Z} [\omega ]}$ .

   To divide an Eisenstein integer ${\displaystyle a+b\omega }$ by another Eisenstein integer ${\displaystyle c+d\omega }$ , notice that ${\displaystyle (c+d\omega )(c+d)(c+d\omega ^{2})=c^{3}+d^{3}}$ ; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by ${\displaystyle (c+d)(c+d\omega ^{2})}$ , then simplify.

• quadratic integer
  • algebraic integer

• Eisenstein prime

• Gaussian integer