"perfect field" meaning in English

{
  "forms": [
    {
      "form": "perfect fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "perfect field (plural perfect fields)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "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": "1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, Addison-Wesley, page 10,\nIf K is a perfect field of prime characteristic p, and if n is a nonnegative integer, then the mapping α→α from K to K is an automorphism."
        },
        {
          "ref": "2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116",
          "text": "So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.",
          "type": "quotation"
        },
        {
          "ref": "2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57, Definition 3.1.7. One says a field K is perfect if any irreducible polynomial in K[X] has as many distinct roots in an algebraic closure as its degree. By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent",
          "text": "a) K is a perfect field;\nb) any irreducible polynomial of K[X] is separable;\nc) any element of an algebraic closure of K is separable over K;\nd) any algebraic extension of K is separable;\ne) for any finite extension K→L, the number of K-homomrphisms from K to an algebraically closed extension of K is equal to [L:K].\nCorollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field."
        }
      ],
      "glosses": [
        "A field K such that every irreducible polynomial over K has distinct roots."
      ],
      "hyponyms": [
        {
          "word": "Galois field"
        }
      ],
      "id": "en-perfect_field-en-noun-srAeNW8~",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "irreducible",
          "irreducible"
        ],
        [
          "polynomial",
          "polynomial"
        ],
        [
          "root",
          "root"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "field such that every irreducible polynomial over it has distinct roots",
          "tags": [
            "masculine"
          ],
          "word": "corps parfait"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "field such that every irreducible polynomial over it has distinct roots",
          "tags": [
            "neuter"
          ],
          "word": "perfekte Körper"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "field such that every irreducible polynomial over it has distinct roots",
          "tags": [
            "neuter"
          ],
          "word": "vollkommene Körper"
        }
      ],
      "wikipedia": [
        "perfect field"
      ]
    }
  ],
  "word": "perfect field"
}

{
  "forms": [
    {
      "form": "perfect fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "perfect field (plural perfect fields)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "Galois field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, Addison-Wesley, page 10,\nIf K is a perfect field of prime characteristic p, and if n is a nonnegative integer, then the mapping α→α from K to K is an automorphism."
        },
        {
          "ref": "2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116",
          "text": "So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.",
          "type": "quotation"
        },
        {
          "ref": "2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57, Definition 3.1.7. One says a field K is perfect if any irreducible polynomial in K[X] has as many distinct roots in an algebraic closure as its degree. By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent",
          "text": "a) K is a perfect field;\nb) any irreducible polynomial of K[X] is separable;\nc) any element of an algebraic closure of K is separable over K;\nd) any algebraic extension of K is separable;\ne) for any finite extension K→L, the number of K-homomrphisms from K to an algebraically closed extension of K is equal to [L:K].\nCorollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field."
        }
      ],
      "glosses": [
        "A field K such that every irreducible polynomial over K has distinct roots."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "field",
          "field"
        ],
        [
          "irreducible",
          "irreducible"
        ],
        [
          "polynomial",
          "polynomial"
        ],
        [
          "root",
          "root"
        ]
      ],
      "qualifier": "field theory",
      "raw_glosses": [
        "(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "perfect field"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "field such that every irreducible polynomial over it has distinct roots",
      "tags": [
        "masculine"
      ],
      "word": "corps parfait"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "field such that every irreducible polynomial over it has distinct roots",
      "tags": [
        "neuter"
      ],
      "word": "perfekte Körper"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "field such that every irreducible polynomial over it has distinct roots",
      "tags": [
        "neuter"
      ],
      "word": "vollkommene Körper"
    }
  ],
  "word": "perfect field"
}