See algebraic number in All languages combined, or Wiktionary
Download JSON data for algebraic number meaning in English (7.5kB)
{
"forms": [
{
"form": "algebraic numbers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic number (plural algebraic numbers)",
"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": "other",
"name": "English entries with topic categories using raw markup",
"parents": [
"Entries with topic categories using raw markup",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "English terms with non-redundant non-automated sortkeys",
"parents": [
"Terms with non-redundant non-automated sortkeys",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Mandarin terms with non-redundant manual transliterations",
"parents": [
"Terms with non-redundant manual transliterations",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Numbers",
"orig": "en:Numbers",
"parents": [
"All topics",
"Terms by semantic function",
"Fundamental"
],
"source": "w"
}
],
"coordinate_terms": [
{
"word": "transcendental number"
}
],
"derived": [
{
"word": "algebraic number theory"
}
],
"examples": [
{
"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."
},
{
"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."
},
{
"ref": "1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435",
"text": "Thus, the equation x-eʸ#x3D;0 is satisfied for x#x3D;1,#x5C;y#x3D;0 and for no other pair of algebraic numbers.",
"type": "quotation"
},
{
"ref": "1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16",
"text": "As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.",
"type": "quotation"
},
{
"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)."
}
],
"glosses": [
"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."
],
"hyponyms": [
{
"sense": "algebraic integer",
"word": "phi"
},
{
"sense": "algebraic integer",
"word": "golden ratio"
}
],
"id": "en-algebraic_number-en-noun-OeOmwQXs",
"links": [
[
"algebra",
"algebra"
],
[
"number theory",
"number theory"
],
[
"complex number",
"complex number"
],
[
"number field",
"number field"
],
[
"root",
"root"
],
[
"polynomial",
"polynomial"
],
[
"coefficient",
"coefficient"
],
[
"integer",
"integer"
],
[
"monic",
"monic"
],
[
"rational number",
"rational number"
]
],
"raw_glosses": [
"(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."
],
"related": [
{
"word": "algebraic integer"
}
],
"topics": [
"algebra",
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"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"
},
{
"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"
},
{
"code": "cs",
"lang": "Czech",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"tags": [
"neuter"
],
"word": "algebraické číslo"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"word": "algebrallinen luku"
},
{
"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"
],
"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"
],
"word": "algebarski broj"
},
{
"code": "sv",
"lang": "Swedish",
"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",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"word": "จำนวนเชิงพีชคณิต"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"tags": [
"plural"
],
"word": "cebirsel sayılar"
}
],
"wikipedia": [
"algebraic number"
]
}
],
"word": "algebraic number"
}
{
"coordinate_terms": [
{
"word": "transcendental number"
}
],
"derived": [
{
"word": "algebraic number theory"
}
],
"forms": [
{
"form": "algebraic numbers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic number (plural algebraic numbers)",
"name": "en-noun"
}
],
"hyponyms": [
{
"sense": "algebraic integer",
"word": "phi"
},
{
"sense": "algebraic integer",
"word": "golden ratio"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "algebraic integer"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English entries with topic categories using raw markup",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with non-redundant non-automated sortkeys",
"English terms with quotations",
"Mandarin terms with non-redundant manual transliterations",
"Quotation templates to be cleaned",
"en:Algebra",
"en:Number theory",
"en:Numbers"
],
"examples": [
{
"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."
},
{
"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."
},
{
"ref": "1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435",
"text": "Thus, the equation x-eʸ#x3D;0 is satisfied for x#x3D;1,#x5C;y#x3D;0 and for no other pair of algebraic numbers.",
"type": "quotation"
},
{
"ref": "1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16",
"text": "As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.",
"type": "quotation"
},
{
"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)."
}
],
"glosses": [
"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."
],
"links": [
[
"algebra",
"algebra"
],
[
"number theory",
"number theory"
],
[
"complex number",
"complex number"
],
[
"number field",
"number field"
],
[
"root",
"root"
],
[
"polynomial",
"polynomial"
],
[
"coefficient",
"coefficient"
],
[
"integer",
"integer"
],
[
"monic",
"monic"
],
[
"rational number",
"rational number"
]
],
"raw_glosses": [
"(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."
],
"topics": [
"algebra",
"mathematics",
"number-theory",
"sciences"
],
"wikipedia": [
"algebraic number"
]
}
],
"translations": [
{
"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"
},
{
"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"
},
{
"code": "cs",
"lang": "Czech",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"tags": [
"neuter"
],
"word": "algebraické číslo"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"word": "algebrallinen luku"
},
{
"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"
],
"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"
],
"word": "algebarski broj"
},
{
"code": "sv",
"lang": "Swedish",
"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",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"word": "จำนวนเชิงพีชคณิต"
},
{
"code": "tr",
"lang": "Turkish",
"sense": "element of a number field that is a root of a polynomial with integer coefficients",
"tags": [
"plural"
],
"word": "cebirsel sayılar"
}
],
"word": "algebraic number"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-05 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). 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.