"subfield" meaning in All languages combined

{
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      "args": {
        "1": "en",
        "2": "sub",
        "3": "field"
      },
      "expansion": "sub- + field",
      "name": "prefix"
    }
  ],
  "etymology_text": "sub- + field",
  "forms": [
    {
      "form": "subfields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "subfield (plural subfields)",
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "glosses": [
        "A smaller, more specialized area of study or occupation within a larger one"
      ],
      "id": "en-subfield-en-noun-rOoOM2i2",
      "translations": [
        {
          "_dis1": "97 3",
          "code": "de",
          "lang": "German",
          "sense": "specialized area of study",
          "tags": [
            "neuter"
          ],
          "word": "Teilgebiet"
        },
        {
          "_dis1": "97 3",
          "code": "pt",
          "lang": "Portuguese",
          "sense": "specialized area of study",
          "tags": [
            "feminine"
          ],
          "word": "subárea"
        },
        {
          "_dis1": "97 3",
          "code": "pt",
          "lang": "Portuguese",
          "sense": "specialized area of study",
          "tags": [
            "masculine"
          ],
          "word": "subcampo"
        },
        {
          "_dis1": "97 3",
          "code": "pa",
          "lang": "Punjabi",
          "roman": "uppakhetar",
          "sense": "specialized area of study",
          "word": "ਉੱਪਖੇਤਰ"
        }
      ]
    },
    {
      "categories": [
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "_dis": "23 77",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w+disamb"
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        {
          "_dis": "28 72",
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          "name": "English entries with topic categories using raw markup",
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          "source": "w+disamb"
        },
        {
          "_dis": "41 59",
          "kind": "other",
          "name": "English terms prefixed with sub-",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "16 84",
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394",
          "text": "Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form\n(1)#x5C;qquadf(c)#x3D;a#x5F;0#x2B;a#x5F;1c#x2B;a#x5F;2c²#x2B;...#x2B;a#x5F;ncⁿ#x5C;qquad#x5C;qquad#x5C;mbox#x7B;(each#x7D;a#x5F;i#x5C;mbox#x7B;in#x7D;F#x5C;mbox#x7B;).#x7D;\n[...]\nIf f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.",
          "type": "quotation"
        },
        {
          "ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397",
          "text": "We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.\nSince, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A subring of a field, containing the multiplicative identity and closed under inversion."
      ],
      "id": "en-subfield-en-noun-Zqmm0IW1",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "subring",
          "subring"
        ],
        [
          "field",
          "field"
        ],
        [
          "identity",
          "identity"
        ],
        [
          "inversion",
          "inversion"
        ]
      ],
      "raw_glosses": [
        "(algebra) A subring of a field, containing the multiplicative identity and closed under inversion."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "subfield"
}

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    "English nouns",
    "English terms prefixed with sub-",
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  "etymology_templates": [
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        "2": "sub",
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    }
  ],
  "etymology_text": "sub- + field",
  "forms": [
    {
      "form": "subfields",
      "tags": [
        "plural"
      ]
    }
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      "expansion": "subfield (plural subfields)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "glosses": [
        "A smaller, more specialized area of study or occupation within a larger one"
      ]
    },
    {
      "categories": [
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394",
          "text": "Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form\n(1)#x5C;qquadf(c)#x3D;a#x5F;0#x2B;a#x5F;1c#x2B;a#x5F;2c²#x2B;...#x2B;a#x5F;ncⁿ#x5C;qquad#x5C;qquad#x5C;mbox#x7B;(each#x7D;a#x5F;i#x5C;mbox#x7B;in#x7D;F#x5C;mbox#x7B;).#x7D;\n[...]\nIf f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.",
          "type": "quotation"
        },
        {
          "ref": "1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397",
          "text": "We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u]. [...] This shows that F[u] is a subfield of K.\nSince, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A subring of a field, containing the multiplicative identity and closed under inversion."
      ],
      "links": [
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          "algebra",
          "algebra"
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        [
          "subring",
          "subring"
        ],
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          "field",
          "field"
        ],
        [
          "identity",
          "identity"
        ],
        [
          "inversion",
          "inversion"
        ]
      ],
      "raw_glosses": [
        "(algebra) A subring of a field, containing the multiplicative identity and closed under inversion."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "specialized area of study",
      "tags": [
        "neuter"
      ],
      "word": "Teilgebiet"
    },
    {
      "code": "pt",
      "lang": "Portuguese",
      "sense": "specialized area of study",
      "tags": [
        "feminine"
      ],
      "word": "subárea"
    },
    {
      "code": "pt",
      "lang": "Portuguese",
      "sense": "specialized area of study",
      "tags": [
        "masculine"
      ],
      "word": "subcampo"
    },
    {
      "code": "pa",
      "lang": "Punjabi",
      "roman": "uppakhetar",
      "sense": "specialized area of study",
      "word": "ਉੱਪਖੇਤਰ"
    }
  ],
  "word": "subfield"
}