"Eisenstein integer" meaning in English

{
  "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"
}