"transcendence degree" meaning in English

{
  "forms": [
    {
      "form": "transcendence degrees",
      "tags": [
        "plural"
      ]
    }
  ],
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      "expansion": "transcendence degree (plural transcendence degrees)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
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          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
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            "Formal sciences",
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      "examples": [
        {
          "text": "2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,\nLet F_1/k and F_2/k denote algebraic function fields of transcendence degree one."
        },
        {
          "ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422",
          "text": "Lemma 7.19 Suppose that L is a field extension of transcendence degree r over a field K and that L is not separably generated over K. If x#x5F;1,#x5C;dots,x#x5F;n are elements of L such that L#x3D;K(x#x5F;1,#x5C;dots,x#x5F;n), then for a suitable relabeling of the x#x5F;i's, the subfield K(x#x5F;1,#x5C;dots,x#x5F;#x7B;r#x2B;1#x7D;) of L is of transcendence degree r and is not separably generated over K.",
          "type": "quotation"
        },
        {
          "ref": "2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24",
          "text": "Proposition 2.1.1 Every finite matrix group #x5C;Gamma#x5C;subset#x5C;operatorname#x7B;GL#x7D;(#x5C;Cⁿ) has n algebraically independent invariants, i.e., the ring #x5C;C#x5B;x#x5D;#x5C;Gamma has transcendence degree n over #x5C;C.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K."
      ],
      "id": "en-transcendence_degree-en-noun-nqwH-2B~",
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          "field extension",
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        ],
        [
          "algebraically independent",
          "algebraically independent"
        ],
        [
          "subset",
          "subset"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K."
      ],
      "raw_tags": [
        "of a field extension"
      ],
      "related": [
        {
          "word": "transcendence"
        },
        {
          "word": "transcendence basis"
        },
        {
          "word": "algebraically independent"
        }
      ],
      "synonyms": [
        {
          "sense": "cardinality of largest algebraically independent subset of a given extension field",
          "word": "transcendental degree"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "transcendence degree"
      ]
    }
  ],
  "word": "transcendence degree"
}

{
  "forms": [
    {
      "form": "transcendence degrees",
      "tags": [
        "plural"
      ]
    }
  ],
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      "expansion": "transcendence degree (plural transcendence degrees)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "transcendence"
    },
    {
      "word": "transcendence basis"
    },
    {
      "word": "algebraically independent"
    }
  ],
  "senses": [
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      "examples": [
        {
          "text": "2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,\nLet F_1/k and F_2/k denote algebraic function fields of transcendence degree one."
        },
        {
          "ref": "2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422",
          "text": "Lemma 7.19 Suppose that L is a field extension of transcendence degree r over a field K and that L is not separably generated over K. If x#x5F;1,#x5C;dots,x#x5F;n are elements of L such that L#x3D;K(x#x5F;1,#x5C;dots,x#x5F;n), then for a suitable relabeling of the x#x5F;i's, the subfield K(x#x5F;1,#x5C;dots,x#x5F;#x7B;r#x2B;1#x7D;) of L is of transcendence degree r and is not separably generated over K.",
          "type": "quotation"
        },
        {
          "ref": "2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24",
          "text": "Proposition 2.1.1 Every finite matrix group #x5C;Gamma#x5C;subset#x5C;operatorname#x7B;GL#x7D;(#x5C;Cⁿ) has n algebraically independent invariants, i.e., the ring #x5C;C#x5B;x#x5D;#x5C;Gamma has transcendence degree n over #x5C;C.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K."
      ],
      "links": [
        [
          "algebra",
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        ],
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          "field extension",
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        ],
        [
          "algebraically independent",
          "algebraically independent"
        ],
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          "subset",
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      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K."
      ],
      "raw_tags": [
        "of a field extension"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "transcendence degree"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "cardinality of largest algebraically independent subset of a given extension field",
      "word": "transcendental degree"
    }
  ],
  "word": "transcendence degree"
}