"algebraic extension" meaning in Anglais

{
  "antonyms": [
    {
      "word": "transcendental extension"
    }
  ],
  "categories": [
    {
      "kind": "other",
      "name": "Locutions nominales en anglais",
      "parents": [],
      "source": "w"
    },
    {
      "kind": "other",
      "name": "Anglais",
      "orig": "anglais",
      "parents": [],
      "source": "w"
    }
  ],
  "forms": [
    {
      "form": "algebraic extensions",
      "tags": [
        "plural"
      ]
    }
  ],
  "hypernyms": [
    {
      "word": "field extension"
    }
  ],
  "hyponyms": [
    {
      "word": "normal extension"
    },
    {
      "word": "separable extension"
    },
    {
      "word": "Galois extension"
    }
  ],
  "lang": "Anglais",
  "lang_code": "en",
  "pos": "noun",
  "pos_title": "Locution nominale",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "Exemples en anglais",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Exemples en anglais à traduire",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Lexique en anglais de l’algèbre",
          "parents": [],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, 1964, page 200",
          "text": "What we now have to prove is that: if an overring R* of R is such that R* is an integral domain, R* is integral over R and the quotient field 𝔏* of R* is a finite algebraic extension of 𝔏, then R* is a local ring and a finite R-module."
        },
        {
          "ref": "Manuel Bronstein, Symbolic Integration I: Transcendental Functions, Springer, 2005",
          "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."
        }
      ],
      "glosses": [
        "Extension algébrique."
      ],
      "id": "fr-algebraic_extension-en-noun-ji2CfGiV",
      "topics": [
        "algebra"
      ]
    }
  ],
  "word": "algebraic extension"
}

{
  "antonyms": [
    {
      "word": "transcendental extension"
    }
  ],
  "categories": [
    "Locutions nominales en anglais",
    "anglais"
  ],
  "forms": [
    {
      "form": "algebraic extensions",
      "tags": [
        "plural"
      ]
    }
  ],
  "hypernyms": [
    {
      "word": "field extension"
    }
  ],
  "hyponyms": [
    {
      "word": "normal extension"
    },
    {
      "word": "separable extension"
    },
    {
      "word": "Galois extension"
    }
  ],
  "lang": "Anglais",
  "lang_code": "en",
  "pos": "noun",
  "pos_title": "Locution nominale",
  "senses": [
    {
      "categories": [
        "Exemples en anglais",
        "Exemples en anglais à traduire",
        "Lexique en anglais de l’algèbre"
      ],
      "examples": [
        {
          "ref": "Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, 1964, page 200",
          "text": "What we now have to prove is that: if an overring R* of R is such that R* is an integral domain, R* is integral over R and the quotient field 𝔏* of R* is a finite algebraic extension of 𝔏, then R* is a local ring and a finite R-module."
        },
        {
          "ref": "Manuel Bronstein, Symbolic Integration I: Transcendental Functions, Springer, 2005",
          "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."
        }
      ],
      "glosses": [
        "Extension algébrique."
      ],
      "topics": [
        "algebra"
      ]
    }
  ],
  "word": "algebraic extension"
}