"extension field" meaning in English

{
  "forms": [
    {
      "form": "extension fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "extension field (plural extension fields)",
      "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": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215",
          "text": "Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).",
          "type": "quotation"
        },
        {
          "ref": "1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56",
          "text": "This extension field of F always contains a root of f in the sense that if K#x3D;F#x5B;x#x5D;#x2F;(f(x)) then x is a root of f(y) in K#x5B;y#x5D;. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.",
          "type": "quotation"
        },
        {
          "ref": "1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53",
          "text": "An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K#x3D;F(#x5C;alpha) if K is the field consisting of all rational expressions formed using #x5C;alpha and elements of F.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements."
      ],
      "hyponyms": [
        {
          "word": "number field"
        },
        {
          "word": "splitting field"
        }
      ],
      "id": "en-extension_field-en-noun-o7ylxBsx",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "subfield",
          "subfield"
        ],
        [
          "base field",
          "base field"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements."
      ],
      "related": [
        {
          "word": "field extension"
        }
      ],
      "synonyms": [
        {
          "english": "where the base field is given",
          "sense": "field that contains a subfield",
          "word": "extension"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "extension field"
      ]
    }
  ],
  "word": "extension field"
}

{
  "forms": [
    {
      "form": "extension fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "extension field (plural extension fields)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "number field"
    },
    {
      "word": "splitting field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "field extension"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215",
          "text": "Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).",
          "type": "quotation"
        },
        {
          "ref": "1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56",
          "text": "This extension field of F always contains a root of f in the sense that if K#x3D;F#x5B;x#x5D;#x2F;(f(x)) then x is a root of f(y) in K#x5B;y#x5D;. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.",
          "type": "quotation"
        },
        {
          "ref": "1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53",
          "text": "An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K#x3D;F(#x5C;alpha) if K is the field consisting of all rational expressions formed using #x5C;alpha and elements of F.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "subfield",
          "subfield"
        ],
        [
          "base field",
          "base field"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "extension field"
      ]
    }
  ],
  "synonyms": [
    {
      "english": "where the base field is given",
      "sense": "field that contains a subfield",
      "word": "extension"
    }
  ],
  "word": "extension field"
}