"algebraic extension" meaning in English

{
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      "form": "algebraic extensions",
      "tags": [
        "plural"
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      "expansion": "algebraic extension (plural algebraic extensions)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w"
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          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
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          "source": "w"
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      "examples": [
        {
          "ref": "1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200",
          "text": "What we now have to prove is that: if an overring R#x2A; of R is such that R#x2A; is an integral domain, R#x2A; is integral over R and the quotient field #x5C;mathfrak#x7B;L#x7D;#x2A; of R#x2A; is a finite algebraic extension of #x5C;mathfrak#x7B;L#x7D;, then R#x2A; is a local ring and a finite R-module.",
          "type": "quotation"
        },
        {
          "ref": "2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110",
          "text": "It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.",
          "type": "quotation"
        },
        {
          "ref": "2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65",
          "text": "We describe the correspondence between subfields of the field P#x5F;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #x5C;pi#x5F;1(X#x5C;setminusO).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
      ],
      "hypernyms": [
        {
          "word": "associative algebra"
        }
      ],
      "hyponyms": [
        {
          "word": "normal extension"
        },
        {
          "word": "separable extension"
        },
        {
          "word": "Galois extension"
        }
      ],
      "id": "en-algebraic_extension-en-noun-LAbntaMK",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field extension",
          "field extension"
        ],
        [
          "algebraic",
          "algebraic"
        ],
        [
          "root",
          "root"
        ],
        [
          "polynomial",
          "polynomial"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "algebraic extension"
      ]
    }
  ],
  "word": "algebraic extension"
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{
  "forms": [
    {
      "form": "algebraic extensions",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "algebraic extension (plural algebraic extensions)",
      "name": "en-noun"
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  "hypernyms": [
    {
      "word": "associative algebra"
    }
  ],
  "hyponyms": [
    {
      "word": "normal extension"
    },
    {
      "word": "separable extension"
    },
    {
      "word": "Galois extension"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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      "examples": [
        {
          "ref": "1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200",
          "text": "What we now have to prove is that: if an overring R#x2A; of R is such that R#x2A; is an integral domain, R#x2A; is integral over R and the quotient field #x5C;mathfrak#x7B;L#x7D;#x2A; of R#x2A; is a finite algebraic extension of #x5C;mathfrak#x7B;L#x7D;, then R#x2A; is a local ring and a finite R-module.",
          "type": "quotation"
        },
        {
          "ref": "2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110",
          "text": "It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.",
          "type": "quotation"
        },
        {
          "ref": "2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65",
          "text": "We describe the correspondence between subfields of the field P#x5F;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #x5C;pi#x5F;1(X#x5C;setminusO).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field extension",
          "field extension"
        ],
        [
          "algebraic",
          "algebraic"
        ],
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          "root",
          "root"
        ],
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          "polynomial",
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      ],
      "qualifier": "field theory",
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        "(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "algebraic extension"
      ]
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  ],
  "word": "algebraic extension"
}