"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"
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          "source": "w"
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        },
        {
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          "name": "Terms with Italian translations",
          "parents": [],
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "The singleton set #92;#123;#92;alpha#92;#125; is algebraically independent over K if and only if the element #92;alpha is transcendental over K.",
          "type": "example"
        },
        {
          "text": "A subset S#92;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#95;i#61;y#95;#123;i,1#125; (where 1 denotes the identity of the semigroup T) we obtain a #92;sigma-algebraically independent over R set #92;#123;y#95;i#92;verti#92;inI#92;#125; such that S#61;R#92;#123;(y#95;i)#95;#123;i#92;inI#125;#92;#125;.",
          "type": "quote"
        },
        {
          "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": [
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      },
      "expansion": "algebraically independent (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
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        "English adjectives",
        "English adverb-adjective phrases",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English terms with quotations",
        "English terms with usage examples",
        "English uncomparable adjectives",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with Italian translations",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "The singleton set #92;#123;#92;alpha#92;#125; is algebraically independent over K if and only if the element #92;alpha is transcendental over K.",
          "type": "example"
        },
        {
          "text": "A subset S#92;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#95;i#61;y#95;#123;i,1#125; (where 1 denotes the identity of the semigroup T) we obtain a #92;sigma-algebraically independent over R set #92;#123;y#95;i#92;verti#92;inI#92;#125; such that S#61;R#92;#123;(y#95;i)#95;#123;i#92;inI#125;#92;#125;.",
          "type": "quote"
        },
        {
          "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"
}