"algebraic number" meaning in All languages combined

See algebraic number on Wiktionary

Noun [English]

Forms: algebraic numbers [plural]
Head templates: {{en-noun}} algebraic number (plural algebraic numbers)
  1. (algebra, number theory) A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers. Wikipedia link: algebraic number Categories (topical): Algebra, Number theory, Numbers Hyponyms (algebraic integer): phi, golden ratio Derived forms: algebraic number theory Related terms: algebraic integer Coordinate_terms: transcendental number Translations (element of a number field that is a root of a polynomial with integer coefficients): 代數數 (Chinese Mandarin), 代数数 (dàishù shù) (Chinese Mandarin), algebraické číslo [neuter] (Czech), algebrallinen luku (Finnish), nombre algébrique [masculine] (French), número alxébrico [masculine] (Galician), algebraische Zahl [feminine] (German), algebrai szám (Hungarian), algebruleg tala [feminine] (Icelandic), algebrutala [feminine] (Icelandic), numero algebrico [masculine] (Italian), алгебралық сан (algebralyq san) (Kazakh), 대수적 수 (daesujeok su) (alt: 代數的數) (Korean), numerus algebraicus [masculine] (Latin), număr algebric [neuter] (Romanian), алгебраическое число (algebraičeskoje čislo) [neuter] (Russian), algebarski broj [masculine] (Serbo-Croatian), algebraiskt tal [neuter] (Swedish), จำนวนเชิงพีชคณิต (jam-nuuan-chəəng-pii-chá-ká-nít) (Thai), cebirsel sayılar [plural] (Turkish)

Inflected forms

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          "text": "The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²+x-1=0, whose coefficients are integers."
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        {
          "text": "The square root of a rational number, √, is an algebraic number since it is a solution of the quadratic equation nx²-m=0, whose coefficients are integers."
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          "ref": "1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:",
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          "ref": "1991, P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83, The existence of such 'transcendental' numbers is well known and it can be proved at three levels",
          "text": "(i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor)."
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        "A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers."
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          "code": "cmn",
          "lang": "Chinese Mandarin",
          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
          "word": "代數數"
        },
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          "lang": "Chinese Mandarin",
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          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
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          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
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        },
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        },
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          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
          "word": "algebrai szám"
        },
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          "lang": "Icelandic",
          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
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          "tags": [
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          "word": "numero algebrico"
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          "alt": "代數的數",
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          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
          "word": "대수적 수"
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          "word": "numerus algebraicus"
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          "word": "număr algebric"
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          "roman": "algebraičeskoje čislo",
          "sense": "element of a number field that is a root of a polynomial with integer coefficients",
          "tags": [
            "neuter"
          ],
          "word": "алгебраическое число"
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          "lang": "Serbo-Croatian",
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          "word": "algebarski broj"
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          "word": "algebraiskt tal"
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          "text": "The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation x²+x-1=0, whose coefficients are integers."
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        },
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      "code": "cmn",
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      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "word": "代數數"
    },
    {
      "code": "cmn",
      "lang": "Chinese Mandarin",
      "roman": "dàishù shù",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "word": "代数数"
    },
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      "code": "cs",
      "lang": "Czech",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "neuter"
      ],
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    },
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      "code": "fi",
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      "word": "algebrallinen luku"
    },
    {
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      "lang": "French",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
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    },
    {
      "code": "gl",
      "lang": "Galician",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "masculine"
      ],
      "word": "número alxébrico"
    },
    {
      "code": "de",
      "lang": "German",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "feminine"
      ],
      "word": "algebraische Zahl"
    },
    {
      "code": "hu",
      "lang": "Hungarian",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "word": "algebrai szám"
    },
    {
      "code": "is",
      "lang": "Icelandic",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "feminine"
      ],
      "word": "algebruleg tala"
    },
    {
      "code": "is",
      "lang": "Icelandic",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "feminine"
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      "word": "algebrutala"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "masculine"
      ],
      "word": "numero algebrico"
    },
    {
      "code": "kk",
      "lang": "Kazakh",
      "roman": "algebralyq san",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "word": "алгебралық сан"
    },
    {
      "alt": "代數的數",
      "code": "ko",
      "lang": "Korean",
      "roman": "daesujeok su",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "word": "대수적 수"
    },
    {
      "code": "la",
      "lang": "Latin",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "masculine"
      ],
      "word": "numerus algebraicus"
    },
    {
      "code": "ro",
      "lang": "Romanian",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "neuter"
      ],
      "word": "număr algebric"
    },
    {
      "code": "ru",
      "lang": "Russian",
      "roman": "algebraičeskoje čislo",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "neuter"
      ],
      "word": "алгебраическое число"
    },
    {
      "code": "sh",
      "lang": "Serbo-Croatian",
      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "masculine"
      ],
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    },
    {
      "code": "sv",
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      "sense": "element of a number field that is a root of a polynomial with integer coefficients",
      "tags": [
        "neuter"
      ],
      "word": "algebraiskt tal"
    },
    {
      "code": "th",
      "lang": "Thai",
      "roman": "jam-nuuan-chəəng-pii-chá-ká-nít",
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      "word": "จำนวนเชิงพีชคณิต"
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      "code": "tr",
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  "word": "algebraic number"
}

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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-15 from the enwiktionary dump dated 2024-12-04 using wiktextract (8a39820 and 4401a4c). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.

If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.