"quadratic field" meaning in All languages combined

{
  "forms": [
    {
      "form": "quadratic fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "quadratic field (plural quadratic fields)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
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          "source": "w"
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      "examples": [
        {
          "text": "1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85: European Conference on Computer Algebra, Proceedings, Volume 2, Springer, LNCS 204, page 279,\nIn a quadratic field mathbf Q(√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known."
        },
        {
          "ref": "1990, Alan Baker, Transcendental Number Theory, Cambridge University Press, page 47",
          "text": "The foundations of the theory of binary quadratic forms, the forerunner of our modern theory of quadratic fields, were laid by Gauss in his famous Disquisitiones Arithmeticae.",
          "type": "quotation"
        },
        {
          "ref": "2000, Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, page 223",
          "text": "In this chapter, we consider the simplest of all number fields that are different from #x5C;Q, i.e. quadratic fields. Since n#x3D;2#x3D;r#x5F;1#x2B;2r#x5F;2, the signature (r#x5F;1,r#x5F;2) of a quadratic field K is either (2,0), in which case we will speak of real quadratic fields, or (0,1), in which case we will speak of imaginary (or complex) quadratic fields. By Proposition 4.8.11 we know that imaginary quadratic fields are those of negative discriminant, and that real quadratic fields are those with positive discriminant.",
          "type": "quotation"
        },
        {
          "text": "2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet, American Mathematical Society, Clay Mathematics Institute, page 247,\nSince Dedekind's time, these conjectures have been phrased in the language of quadratic fields. […] Throughout this paper, k= Q (√d) will be a quadratic field of discriminant d and h(k) or sometimes h(d) will be the class-number of k."
        }
      ],
      "glosses": [
        "A number field that is an extension field of degree two over the rational numbers."
      ],
      "hypernyms": [
        {
          "word": "number field"
        }
      ],
      "hyponyms": [
        {
          "word": "complex quadratic field"
        },
        {
          "word": "imaginary quadratic field"
        },
        {
          "word": "real quadratic field"
        }
      ],
      "id": "en-quadratic_field-en-noun-3Xo0w9UO",
      "links": [
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        [
          "extension field",
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        ],
        [
          "degree",
          "degree"
        ],
        [
          "rational numbers",
          "rational numbers"
        ]
      ],
      "qualifier": "algebraic number theory",
      "raw_glosses": [
        "(algebraic number theory) A number field that is an extension field of degree two over the rational numbers."
      ],
      "related": [
        {
          "word": "quadratic integer"
        },
        {
          "word": "binary quadratic form"
        },
        {
          "word": "quadratic form"
        }
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "number field that is an extension field of order 2 over the rational numbers",
          "tags": [
            "masculine"
          ],
          "word": "quadratischer Zahlkörper"
        }
      ],
      "wikipedia": [
        "quadratic field"
      ]
    }
  ],
  "word": "quadratic field"
}

{
  "forms": [
    {
      "form": "quadratic fields",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "quadratic field (plural quadratic fields)",
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  "hypernyms": [
    {
      "word": "number field"
    }
  ],
  "hyponyms": [
    {
      "word": "complex quadratic field"
    },
    {
      "word": "imaginary quadratic field"
    },
    {
      "word": "real quadratic field"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "quadratic integer"
    },
    {
      "word": "binary quadratic form"
    },
    {
      "word": "quadratic form"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations"
      ],
      "examples": [
        {
          "text": "1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85: European Conference on Computer Algebra, Proceedings, Volume 2, Springer, LNCS 204, page 279,\nIn a quadratic field mathbf Q(√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known."
        },
        {
          "ref": "1990, Alan Baker, Transcendental Number Theory, Cambridge University Press, page 47",
          "text": "The foundations of the theory of binary quadratic forms, the forerunner of our modern theory of quadratic fields, were laid by Gauss in his famous Disquisitiones Arithmeticae.",
          "type": "quotation"
        },
        {
          "ref": "2000, Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, page 223",
          "text": "In this chapter, we consider the simplest of all number fields that are different from #x5C;Q, i.e. quadratic fields. Since n#x3D;2#x3D;r#x5F;1#x2B;2r#x5F;2, the signature (r#x5F;1,r#x5F;2) of a quadratic field K is either (2,0), in which case we will speak of real quadratic fields, or (0,1), in which case we will speak of imaginary (or complex) quadratic fields. By Proposition 4.8.11 we know that imaginary quadratic fields are those of negative discriminant, and that real quadratic fields are those with positive discriminant.",
          "type": "quotation"
        },
        {
          "text": "2007, H. M. Stark, The Gauss Class-Number Problems, William Duke, Yuri Tschinkel (editors), Analytic Number Theory: A Tribute to Gauss and Dirichlet, American Mathematical Society, Clay Mathematics Institute, page 247,\nSince Dedekind's time, these conjectures have been phrased in the language of quadratic fields. […] Throughout this paper, k= Q (√d) will be a quadratic field of discriminant d and h(k) or sometimes h(d) will be the class-number of k."
        }
      ],
      "glosses": [
        "A number field that is an extension field of degree two over the rational numbers."
      ],
      "links": [
        [
          "number field",
          "number field"
        ],
        [
          "extension field",
          "extension field"
        ],
        [
          "degree",
          "degree"
        ],
        [
          "rational numbers",
          "rational numbers"
        ]
      ],
      "qualifier": "algebraic number theory",
      "raw_glosses": [
        "(algebraic number theory) A number field that is an extension field of degree two over the rational numbers."
      ],
      "wikipedia": [
        "quadratic field"
      ]
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "number field that is an extension field of order 2 over the rational numbers",
      "tags": [
        "masculine"
      ],
      "word": "quadratischer Zahlkörper"
    }
  ],
  "word": "quadratic field"
}