"algebraically independent" meaning in English

{
  "antonyms": [
    {
      "sense": "antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”",
      "word": "algebraically dependent"
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "algebraically independent (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "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"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "The singleton set #x5C;#x7B;#x5C;alpha#x5C;#x7D; is algebraically independent over K if and only if the element #x5C;alpha is transcendental over K.",
          "type": "example"
        },
        {
          "text": "A subset S#x5C;subsetL is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.",
          "type": "example"
        },
        {
          "text": "1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40,\nIf the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0."
        },
        {
          "ref": "2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251",
          "text": "Setting y#x5F;i#x3D;y#x5F;#x7B;i,1#x7D; (where 1 denotes the identity of the semigroup T) we obtain a #x5C;sigma-algebraically independent over R set #x5C;#x7B;y#x5F;i#x5C;verti#x5C;inI#x5C;#x7D; such that S#x3D;R#x5C;#x7B;(y#x5F;i)#x5F;#x7B;i#x5C;inI#x7D;#x5C;#x7D;.",
          "type": "quotation"
        },
        {
          "ref": "2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138, Let us begin with the following conjecture of Schneider",
          "text": "If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then\nαᵝ,…,α\nare algebraically independent."
        }
      ],
      "glosses": [
        "(Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K."
      ],
      "id": "en-algebraically_independent-en-adj-8jru25mh",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "extension field",
          "extension field"
        ],
        [
          "field extension",
          "field extension"
        ],
        [
          "polynomial equation",
          "polynomial equation"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) (Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K."
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field",
          "tags": [
            "feminine",
            "masculine"
          ],
          "word": "algebricamente indipendente"
        }
      ]
    }
  ],
  "word": "algebraically independent"
}

{
  "antonyms": [
    {
      "sense": "antonym(s) of “which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field”",
      "word": "algebraically dependent"
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "algebraically independent (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
      "categories": [
        "English adjectives",
        "English adverb-adjective phrases",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English lemmas",
        "English multiword terms",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "English terms with usage examples",
        "English uncomparable adjectives",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "The singleton set #x5C;#x7B;#x5C;alpha#x5C;#x7D; is algebraically independent over K if and only if the element #x5C;alpha is transcendental over K.",
          "type": "example"
        },
        {
          "text": "A subset S#x5C;subsetL is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.",
          "type": "example"
        },
        {
          "text": "1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40,\nIf the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0."
        },
        {
          "ref": "2006, Alexander B. Levin, “Difference algebra”, in M. Hazewinkel, editor, Handbook of Algebra, Volume 4, Elsevier (North-Holland), page 251",
          "text": "Setting y#x5F;i#x3D;y#x5F;#x7B;i,1#x7D; (where 1 denotes the identity of the semigroup T) we obtain a #x5C;sigma-algebraically independent over R set #x5C;#x7B;y#x5F;i#x5C;verti#x5C;inI#x5C;#x7D; such that S#x3D;R#x5C;#x7B;(y#x5F;i)#x5F;#x7B;i#x5C;inI#x7D;#x5C;#x7D;.",
          "type": "quotation"
        },
        {
          "ref": "2014, M. Ram Murty, Purusottam Rath, Transcendental Numbers, Springer, page 138, Let us begin with the following conjecture of Schneider",
          "text": "If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then\nαᵝ,…,α\nare algebraically independent."
        }
      ],
      "glosses": [
        "(Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "subset",
          "subset"
        ],
        [
          "extension field",
          "extension field"
        ],
        [
          "field extension",
          "field extension"
        ],
        [
          "polynomial equation",
          "polynomial equation"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) (Of a subset S of the extension field L of a given field extension L / K) whose elements do not satisfy any non-trivial polynomial equation with coefficients in K."
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "which does not or whose elements do not satisfy any nontrivial polynomial equation over a given field",
      "tags": [
        "feminine",
        "masculine"
      ],
      "word": "algebricamente indipendente"
    }
  ],
  "word": "algebraically independent"
}