"Galois group" meaning in English

{
  "etymology_text": "Named after Évariste Galois, who first discovered them.",
  "forms": [
    {
      "form": "Galois groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Galois group (plural Galois groups)",
      "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": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
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          "source": "w"
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          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1996, Patrick Morandi, Field and Galois Theory, Springer, page 123",
          "text": "In this section, we show how to determine the Galois group and the roots of an irreducible polynomial of degree 2, 3, or 4.",
          "type": "quotation"
        },
        {
          "ref": "2004, George Szeto, Liangyong Xue, “On Central Galois Algebras of a Galois Algebra”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, CRC Press, page 493",
          "text": "Let B be a Galois algebra over R with Galois group G, C the center of B, and K#x3D;#x5C;#x7B;g#x5C;inG#x5C;vertg(c)#x3D;c#x5C;#x5C;mathsf#x7B;for#x5C;each#x7D;#x5C;c#x5C;inC#x5C;#x7D;. A natural question is whether B is a central Galois algebra with Galois group K.",
          "type": "quotation"
        },
        {
          "ref": "2009, Steven H. Weintraub, Galois Theory, 2nd edition, Springer, page 128",
          "text": "Our work actually gives an algorithm for computing Galois groups of polynomials f(X)#x5C;in#x5C;Q#x5B;X#x5D;:",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The automorphism group of a Galois extension."
      ],
      "id": "en-Galois_group-en-noun-WBEU9lYb",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "automorphism group",
          "automorphism group"
        ],
        [
          "Galois extension",
          "Galois extension"
        ]
      ],
      "qualifier": "Galois theory",
      "raw_glosses": [
        "(algebra, Galois theory) The automorphism group of a Galois extension."
      ],
      "related": [
        {
          "word": "Galois algebra"
        },
        {
          "word": "Galois closure"
        },
        {
          "word": "Galois extension"
        },
        {
          "word": "Galois theory"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "ar",
          "lang": "Arabic",
          "sense": "specific group associated with a field extension",
          "word": "زمرة غالوا"
        },
        {
          "code": "cmn",
          "lang": "Chinese Mandarin",
          "sense": "specific group associated with a field extension",
          "word": "伽罗瓦群"
        },
        {
          "code": "fr",
          "lang": "French",
          "sense": "specific group associated with a field extension",
          "word": "groupe de Galois"
        },
        {
          "code": "hu",
          "lang": "Hungarian",
          "sense": "specific group associated with a field extension",
          "word": "Galois-csoport"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "specific group associated with a field extension",
          "tags": [
            "masculine"
          ],
          "word": "gruppo di Galois"
        },
        {
          "code": "ja",
          "lang": "Japanese",
          "sense": "specific group associated with a field extension",
          "word": "ガロア群"
        },
        {
          "code": "ko",
          "lang": "Korean",
          "roman": "gallua gun",
          "sense": "specific group associated with a field extension",
          "word": "갈루아 군"
        },
        {
          "code": "ru",
          "lang": "Russian",
          "roman": "gruppa Galua",
          "sense": "specific group associated with a field extension",
          "word": "группа Галуа"
        },
        {
          "code": "es",
          "lang": "Spanish",
          "sense": "specific group associated with a field extension",
          "word": "grupo de Galois"
        }
      ],
      "wikipedia": [
        "Galois group",
        "Évariste Galois"
      ]
    }
  ],
  "word": "Galois group"
}

{
  "etymology_text": "Named after Évariste Galois, who first discovered them.",
  "forms": [
    {
      "form": "Galois groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Galois group (plural Galois groups)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Galois algebra"
    },
    {
      "word": "Galois closure"
    },
    {
      "word": "Galois extension"
    },
    {
      "word": "Galois theory"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "ref": "1996, Patrick Morandi, Field and Galois Theory, Springer, page 123",
          "text": "In this section, we show how to determine the Galois group and the roots of an irreducible polynomial of degree 2, 3, or 4.",
          "type": "quotation"
        },
        {
          "ref": "2004, George Szeto, Liangyong Xue, “On Central Galois Algebras of a Galois Algebra”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, CRC Press, page 493",
          "text": "Let B be a Galois algebra over R with Galois group G, C the center of B, and K#x3D;#x5C;#x7B;g#x5C;inG#x5C;vertg(c)#x3D;c#x5C;#x5C;mathsf#x7B;for#x5C;each#x7D;#x5C;c#x5C;inC#x5C;#x7D;. A natural question is whether B is a central Galois algebra with Galois group K.",
          "type": "quotation"
        },
        {
          "ref": "2009, Steven H. Weintraub, Galois Theory, 2nd edition, Springer, page 128",
          "text": "Our work actually gives an algorithm for computing Galois groups of polynomials f(X)#x5C;in#x5C;Q#x5B;X#x5D;:",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The automorphism group of a Galois extension."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "automorphism group",
          "automorphism group"
        ],
        [
          "Galois extension",
          "Galois extension"
        ]
      ],
      "qualifier": "Galois theory",
      "raw_glosses": [
        "(algebra, Galois theory) The automorphism group of a Galois extension."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Galois group",
        "Évariste Galois"
      ]
    }
  ],
  "translations": [
    {
      "code": "ar",
      "lang": "Arabic",
      "sense": "specific group associated with a field extension",
      "word": "زمرة غالوا"
    },
    {
      "code": "cmn",
      "lang": "Chinese Mandarin",
      "sense": "specific group associated with a field extension",
      "word": "伽罗瓦群"
    },
    {
      "code": "fr",
      "lang": "French",
      "sense": "specific group associated with a field extension",
      "word": "groupe de Galois"
    },
    {
      "code": "hu",
      "lang": "Hungarian",
      "sense": "specific group associated with a field extension",
      "word": "Galois-csoport"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "specific group associated with a field extension",
      "tags": [
        "masculine"
      ],
      "word": "gruppo di Galois"
    },
    {
      "code": "ja",
      "lang": "Japanese",
      "sense": "specific group associated with a field extension",
      "word": "ガロア群"
    },
    {
      "code": "ko",
      "lang": "Korean",
      "roman": "gallua gun",
      "sense": "specific group associated with a field extension",
      "word": "갈루아 군"
    },
    {
      "code": "ru",
      "lang": "Russian",
      "roman": "gruppa Galua",
      "sense": "specific group associated with a field extension",
      "word": "группа Галуа"
    },
    {
      "code": "es",
      "lang": "Spanish",
      "sense": "specific group associated with a field extension",
      "word": "grupo de Galois"
    }
  ],
  "word": "Galois group"
}