"separable polynomial" meaning in English

{
  "forms": [
    {
      "form": "separable polynomials",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "separable polynomial (plural separable polynomials)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
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        }
      ],
      "coordinate_terms": [
        {
          "word": "inseparable polynomial"
        }
      ],
      "derived": [
        {
          "word": "discriminantly separable polynomial"
        }
      ],
      "examples": [
        {
          "text": "Over a perfect field, the separable polynomials are precisely the square-free polynomials.",
          "type": "example"
        },
        {
          "text": "The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.",
          "type": "example"
        },
        {
          "ref": "1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:",
          "text": "We know that F(#x5C;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).",
          "type": "quote"
        },
        {
          "ref": "2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:",
          "text": "Proposition 1.4.2 A finite field extension M#x2F;K is Galois if and only if M is the splitting field over K of a separable polynomial.",
          "type": "quote"
        },
        {
          "ref": "2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:",
          "text": "If #x5C;pi#x5F;P is a separable polynomial in K#x5B;t#x5D;, then the derivative #x5C;partial#x5F;t#x5C;pi#x5F;P is prime to #x5C;pi#x5F;P in K#x5B;t#x5D;, and therefore a unit in K#x5B;t#x5D;#x5F;P.[…]In the case when #x5C;pi#x5F;P is an inseparable polynomial we may write #x5C;pi#x5F;P#x3D;f(t#x7B;pʳ#x7D;) for a suitable r#x3E;0 and separable polynomial f.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial)."
      ],
      "id": "en-separable_polynomial-en-noun-szTRRYb8",
      "links": [
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          "algebra",
          "algebra"
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        [
          "polynomial",
          "polynomial"
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        [
          "distinct",
          "distinct"
        ],
        [
          "algebraic closure",
          "algebraic closure"
        ],
        [
          "degree",
          "degree"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial)."
      ],
      "related": [
        {
          "word": "separable extension"
        },
        {
          "word": "splitting field"
        },
        {
          "word": "square-free polynomial"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "polynomial that has distinct roots",
          "tags": [
            "masculine"
          ],
          "word": "polinomio separabile"
        },
        {
          "code": "es",
          "lang": "Spanish",
          "sense": "polynomial that has distinct roots",
          "tags": [
            "masculine"
          ],
          "word": "polinomio separable"
        }
      ],
      "wikipedia": [
        "separable polynomial"
      ]
    }
  ],
  "word": "separable polynomial"
}

{
  "coordinate_terms": [
    {
      "word": "inseparable polynomial"
    }
  ],
  "derived": [
    {
      "word": "discriminantly separable polynomial"
    }
  ],
  "forms": [
    {
      "form": "separable polynomials",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "separable polynomial (plural separable polynomials)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "separable extension"
    },
    {
      "word": "splitting field"
    },
    {
      "word": "square-free polynomial"
    }
  ],
  "senses": [
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        "Terms with Italian translations",
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      "examples": [
        {
          "text": "Over a perfect field, the separable polynomials are precisely the square-free polynomials.",
          "type": "example"
        },
        {
          "text": "The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.",
          "type": "example"
        },
        {
          "ref": "1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:",
          "text": "We know that F(#x5C;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).",
          "type": "quote"
        },
        {
          "ref": "2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:",
          "text": "Proposition 1.4.2 A finite field extension M#x2F;K is Galois if and only if M is the splitting field over K of a separable polynomial.",
          "type": "quote"
        },
        {
          "ref": "2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:",
          "text": "If #x5C;pi#x5F;P is a separable polynomial in K#x5B;t#x5D;, then the derivative #x5C;partial#x5F;t#x5C;pi#x5F;P is prime to #x5C;pi#x5F;P in K#x5B;t#x5D;, and therefore a unit in K#x5B;t#x5D;#x5F;P.[…]In the case when #x5C;pi#x5F;P is an inseparable polynomial we may write #x5C;pi#x5F;P#x3D;f(t#x7B;pʳ#x7D;) for a suitable r#x3E;0 and separable polynomial f.",
          "type": "quote"
        }
      ],
      "glosses": [
        "A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial)."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "polynomial",
          "polynomial"
        ],
        [
          "distinct",
          "distinct"
        ],
        [
          "algebraic closure",
          "algebraic closure"
        ],
        [
          "degree",
          "degree"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial)."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
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      "wikipedia": [
        "separable polynomial"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "polynomial that has distinct roots",
      "tags": [
        "masculine"
      ],
      "word": "polinomio separabile"
    },
    {
      "code": "es",
      "lang": "Spanish",
      "sense": "polynomial that has distinct roots",
      "tags": [
        "masculine"
      ],
      "word": "polinomio separable"
    }
  ],
  "word": "separable polynomial"
}