"algebraic closure" meaning in English

{
  "forms": [
    {
      "form": "algebraic closures",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "algebraic closure (plural algebraic closures)",
      "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"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University Press, page 1",
          "text": "The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.",
          "type": "quotation"
        },
        {
          "ref": "2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127",
          "text": "It turns out that the algebraic closure #x5C;mathbf#x7B;Q#x7D;#x5C;mathrm#x7B;a#x7D;#x5F;p is not complete, so we shall consider its completion #x5C;mathbf#x7B;C#x7D;#x5F;p: This field turns out to be algebraically closed and is a natural domain for the study of \"analytic functions.\"",
          "type": "quotation"
        },
        {
          "ref": "2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179",
          "text": "While #x5C;Complex contains an algebraic closure of #x5C;mathbb#x7B;Q#x7D;, it is by no means the only algebraically closed field containing an algebraic closure of #x5C;mathbb#x7B;Q#x7D;. We denote by #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; the algebraic closure of #x5C;mathbb#x7B;Q#x7D; in #x5C;Complex; this field is simply the subfield of #x5C;Complex consisting of algebraic numbers. The field #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; is isomorphic, then, to any algebraic closure of #x5C;mathbb#x7B;Q#x7D;, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; than we were.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G)."
      ],
      "id": "en-algebraic_closure-en-noun-5Ddx458M",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "root",
          "root"
        ],
        [
          "polynomial",
          "polynomial"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory, of a field F) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G)."
      ],
      "raw_tags": [
        "of a field F"
      ],
      "related": [
        {
          "word": "algebraically closed"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "field which is algebraically closed over another one",
          "word": "algebrallinen sulkeuma"
        },
        {
          "code": "pt",
          "lang": "Portuguese",
          "sense": "field which is algebraically closed over another one",
          "tags": [
            "masculine"
          ],
          "word": "fecho algébrico"
        }
      ],
      "wikipedia": [
        "algebraic closure"
      ]
    }
  ],
  "word": "algebraic closure"
}

{
  "forms": [
    {
      "form": "algebraic closures",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "algebraic closure (plural algebraic closures)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "algebraically closed"
    }
  ],
  "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": "1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University Press, page 1",
          "text": "The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.",
          "type": "quotation"
        },
        {
          "ref": "2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127",
          "text": "It turns out that the algebraic closure #x5C;mathbf#x7B;Q#x7D;#x5C;mathrm#x7B;a#x7D;#x5F;p is not complete, so we shall consider its completion #x5C;mathbf#x7B;C#x7D;#x5F;p: This field turns out to be algebraically closed and is a natural domain for the study of \"analytic functions.\"",
          "type": "quotation"
        },
        {
          "ref": "2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179",
          "text": "While #x5C;Complex contains an algebraic closure of #x5C;mathbb#x7B;Q#x7D;, it is by no means the only algebraically closed field containing an algebraic closure of #x5C;mathbb#x7B;Q#x7D;. We denote by #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; the algebraic closure of #x5C;mathbb#x7B;Q#x7D; in #x5C;Complex; this field is simply the subfield of #x5C;Complex consisting of algebraic numbers. The field #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; is isomorphic, then, to any algebraic closure of #x5C;mathbb#x7B;Q#x7D;, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with #x5C;mathbb#x7B;Q#x7D;#x5C;mathrm#x7B;alg#x7D; than we were.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "root",
          "root"
        ],
        [
          "polynomial",
          "polynomial"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory, of a field F) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G)."
      ],
      "raw_tags": [
        "of a field F"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "algebraic closure"
      ]
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "field which is algebraically closed over another one",
      "word": "algebrallinen sulkeuma"
    },
    {
      "code": "pt",
      "lang": "Portuguese",
      "sense": "field which is algebraically closed over another one",
      "tags": [
        "masculine"
      ],
      "word": "fecho algébrico"
    }
  ],
  "word": "algebraic closure"
}