"cyclotomic field" meaning in All languages combined

{
  "forms": [
    {
      "form": "cyclotomic fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "cyclotomic field (plural cyclotomic 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": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "A cyclotomic field is the splitting field of the cyclotomic polynomial #x5C;Phi#x5F;n(x)#x3D;#x5C;prod#x5F;#x5C;stackrel#x7B;1#x5C;lek#x5C;len#x7D;#x7B;#x5C;gcd(k,n)#x3D;1#x7D;#x5C;left(x-e#x7B;#x5C;frac#x7B;2k#x5C;pii#x7D;#x7B;n) and consequently is a Galois extension of the field of rational numbers.}}",
          "type": "example"
        },
        {
          "ref": "1991, A. Fröhlich, M. J. Taylor, “Algebraic Number Theory”, in Paperback, Cambridge University Press, published 1993, page 205",
          "text": "Cyclotomic fields are fields obtained by adjoining to #x5C;Q roots of unity, i.e. roots of polynomials of the form Xⁿ-1, although the reader is warned that this terminology will be extended in §2.[…]Cyclotomic fields play a fundamental role in a number of arithmetic problems: for instance primes in arithmetic progression (see VIII,§4) and Fermat's Last Theorem (VII,§1).",
          "type": "quotation"
        },
        {
          "ref": "2007, Israel Kleiner, A History of Abstract Algebra, Springer (Birkhäuser), page 26",
          "text": "At about the same time Kummer introduced his \"ideal numbers,\" defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field—in our terminology, the order of the ideal class group of the cyclotomic field.",
          "type": "quotation"
        },
        {
          "ref": "2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 94",
          "text": "As in the case of the Drinfel'd double,¹³¹ it can be shown that these fields are subfields of the cyclotomic field #x5C;Q#x5F;N. Since the Galois group of #x5C;Q#x5F;N is abelian, every subfield of the cyclotomic field is normal, and consequently preserved by the action of the Galois group of #x5C;Q#x5F;N.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A number field obtained by adjoining a primitive root of unity to the field of rational numbers."
      ],
      "hypernyms": [
        {
          "word": "number field"
        }
      ],
      "id": "en-cyclotomic_field-en-noun-GXEyBKVM",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "number field",
          "number field"
        ],
        [
          "primitive",
          "primitive root"
        ],
        [
          "root of unity",
          "root of unity"
        ],
        [
          "rational numbers",
          "rational numbers"
        ]
      ],
      "meronyms": [
        {
          "word": "Kummer ring"
        },
        {
          "word": "cyclotomic integer"
        }
      ],
      "qualifier": "algebraic number theory",
      "raw_glosses": [
        "(number theory, algebraic number theory) A number field obtained by adjoining a primitive root of unity to the field of rational numbers."
      ],
      "related": [
        {
          "word": "cyclotomic polynomial"
        },
        {
          "word": "quadratic field"
        }
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
          "tags": [
            "masculine"
          ],
          "word": "Kreisteilungskörper"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
          "tags": [
            "masculine"
          ],
          "word": "zyklotomische Körper"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
          "tags": [
            "masculine"
          ],
          "word": "campo ciclotomico"
        },
        {
          "code": "es",
          "lang": "Spanish",
          "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
          "tags": [
            "masculine"
          ],
          "word": "cuerpo ciclotómico"
        }
      ],
      "wikipedia": [
        "cyclotomic field"
      ]
    }
  ],
  "word": "cyclotomic field"
}

{
  "forms": [
    {
      "form": "cyclotomic fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "cyclotomic field (plural cyclotomic fields)",
      "name": "en-noun"
    }
  ],
  "hypernyms": [
    {
      "word": "number field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "meronyms": [
    {
      "word": "Kummer ring"
    },
    {
      "word": "cyclotomic integer"
    }
  ],
  "pos": "noun",
  "related": [
    {
      "word": "cyclotomic polynomial"
    },
    {
      "word": "quadratic field"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "English terms with usage examples",
        "en:Number theory"
      ],
      "examples": [
        {
          "text": "A cyclotomic field is the splitting field of the cyclotomic polynomial #x5C;Phi#x5F;n(x)#x3D;#x5C;prod#x5F;#x5C;stackrel#x7B;1#x5C;lek#x5C;len#x7D;#x7B;#x5C;gcd(k,n)#x3D;1#x7D;#x5C;left(x-e#x7B;#x5C;frac#x7B;2k#x5C;pii#x7D;#x7B;n) and consequently is a Galois extension of the field of rational numbers.}}",
          "type": "example"
        },
        {
          "ref": "1991, A. Fröhlich, M. J. Taylor, “Algebraic Number Theory”, in Paperback, Cambridge University Press, published 1993, page 205",
          "text": "Cyclotomic fields are fields obtained by adjoining to #x5C;Q roots of unity, i.e. roots of polynomials of the form Xⁿ-1, although the reader is warned that this terminology will be extended in §2.[…]Cyclotomic fields play a fundamental role in a number of arithmetic problems: for instance primes in arithmetic progression (see VIII,§4) and Fermat's Last Theorem (VII,§1).",
          "type": "quotation"
        },
        {
          "ref": "2007, Israel Kleiner, A History of Abstract Algebra, Springer (Birkhäuser), page 26",
          "text": "At about the same time Kummer introduced his \"ideal numbers,\" defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field—in our terminology, the order of the ideal class group of the cyclotomic field.",
          "type": "quotation"
        },
        {
          "ref": "2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 94",
          "text": "As in the case of the Drinfel'd double,¹³¹ it can be shown that these fields are subfields of the cyclotomic field #x5C;Q#x5F;N. Since the Galois group of #x5C;Q#x5F;N is abelian, every subfield of the cyclotomic field is normal, and consequently preserved by the action of the Galois group of #x5C;Q#x5F;N.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A number field obtained by adjoining a primitive root of unity to the field of rational numbers."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "number field",
          "number field"
        ],
        [
          "primitive",
          "primitive root"
        ],
        [
          "root of unity",
          "root of unity"
        ],
        [
          "rational numbers",
          "rational numbers"
        ]
      ],
      "qualifier": "algebraic number theory",
      "raw_glosses": [
        "(number theory, algebraic number theory) A number field obtained by adjoining a primitive root of unity to the field of rational numbers."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "cyclotomic field"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
      "tags": [
        "masculine"
      ],
      "word": "Kreisteilungskörper"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
      "tags": [
        "masculine"
      ],
      "word": "zyklotomische Körper"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
      "tags": [
        "masculine"
      ],
      "word": "campo ciclotomico"
    },
    {
      "code": "es",
      "lang": "Spanish",
      "sense": "extension field of the rational numbers formed by adjoining a primitive root of unity",
      "tags": [
        "masculine"
      ],
      "word": "cuerpo ciclotómico"
    }
  ],
  "word": "cyclotomic field"
}