See Eisenstein integer in All languages combined, or Wiktionary
{ "etymology_text": "Named after Gotthold Eisenstein (1823–1852), German mathematician.", "forms": [ { "form": "Eisenstein integers", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "Eisenstein integer (plural Eisenstein integers)", "name": "en-noun" } ], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [ { "categories": [ { "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w" }, { "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w" }, { "kind": "other", "name": "Pages with entries", "parents": [], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Numbers", "orig": "en:Numbers", "parents": [ "All topics", "Terms by semantic function", "Fundamental" ], "source": "w" } ], "examples": [ { "text": "To divide an Eisenstein integer a#x2B;b#x5C;omega by another Eisenstein integer c#x2B;d#x5C;omega, notice that (c#x2B;d#x5C;omega)(c#x2B;d)(c#x2B;d#x5C;omega²)#x3D;c³#x2B;d³; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by (c#x2B;d)(c#x2B;d#x5C;omega²), then simplify.", "type": "example" } ], "glosses": [ "A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω]." ], "hypernyms": [ { "sense": "quadratic integer", "word": "algebraic integer" } ], "hyponyms": [ { "word": "Eisenstein prime" } ], "id": "en-Eisenstein_integer-en-noun-FYHye8C5", "links": [ [ "algebra", "algebra" ], [ "complex number", "complex number" ], [ "integer", "integer" ], [ "Euclidean domain", "Euclidean domain" ] ], "raw_glosses": [ "(algebra) A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω]." ], "related": [ { "word": "Gaussian integer" } ], "topics": [ "algebra", "mathematics", "sciences" ], "wikipedia": [ "Eisenstein integer", "Gotthold Eisenstein" ] } ], "word": "Eisenstein integer" }
{ "etymology_text": "Named after Gotthold Eisenstein (1823–1852), German mathematician.", "forms": [ { "form": "Eisenstein integers", "tags": [ "plural" ] } ], "head_templates": [ { "args": {}, "expansion": "Eisenstein integer (plural Eisenstein integers)", "name": "en-noun" } ], "hypernyms": [ { "sense": "quadratic integer", "word": "algebraic integer" } ], "hyponyms": [ { "word": "Eisenstein prime" } ], "lang": "English", "lang_code": "en", "pos": "noun", "related": [ { "word": "Gaussian integer" } ], "senses": [ { "categories": [ "English countable nouns", "English entries with incorrect language header", "English eponyms", "English lemmas", "English multiword terms", "English nouns", "English terms with usage examples", "Pages with 1 entry", "Pages with entries", "en:Algebra", "en:Numbers" ], "examples": [ { "text": "To divide an Eisenstein integer a#x2B;b#x5C;omega by another Eisenstein integer c#x2B;d#x5C;omega, notice that (c#x2B;d#x5C;omega)(c#x2B;d)(c#x2B;d#x5C;omega²)#x3D;c³#x2B;d³; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by (c#x2B;d)(c#x2B;d#x5C;omega²), then simplify.", "type": "example" } ], "glosses": [ "A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω]." ], "links": [ [ "algebra", "algebra" ], [ "complex number", "complex number" ], [ "integer", "integer" ], [ "Euclidean domain", "Euclidean domain" ] ], "raw_glosses": [ "(algebra) A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω]." ], "topics": [ "algebra", "mathematics", "sciences" ], "wikipedia": [ "Eisenstein integer", "Gotthold Eisenstein" ] } ], "word": "Eisenstein integer" }
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