Eisenstein integer

English

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

Eisenstein integer (plural Eisenstein integers)

(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.

单词	Eisenstein integer
释义	Eisenstein integer English Etymology Named after Gotthold Eisenstein (1823–1852), German mathematician. Noun Eisenstein integer (plural Eisenstein integers) (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. Hypernyms quadratic integer algebraic integer Hyponyms Eisenstein prime See also Gaussian integer
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