See algebraic integer in All languages combined, or Wiktionary
Download JSON data for algebraic integer meaning in English (5.8kB)
{
"forms": [
{
"form": "algebraic integers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic integer (plural algebraic integers)",
"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 redundant transliterations",
"parents": [
"Terms with redundant 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"
}
],
"examples": [
{
"text": "A Gaussian integer z=a+ib is an algebraic integer since it is a solution of either the equation z²+(-2a)z+(a²+b²)=0 or the equation z-a=0."
},
{
"ref": "1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62",
"text": "An algebraic number is said to be an algebraic integer if the coefficient of the highest power of x in the minimal polynomial P is 1. The algebraic integers in an algebraic number field k form a ring R.",
"type": "quotation"
},
{
"text": "1989, Heinrich Rolletschek, Shortest Division Chains in Imaginary Quadratic Number Fields, Patrizia Gianni (editor), Symbolic and Algebraic Computation: International Symposium, Springer, LNCS 358, page 231,\nLet O_d be the set of algebraic integers in an imaginary quadratic number field Q [√],d<0, where d is the discriminant of O_d."
},
{
"ref": "2010, Pierre Moussa, “Localisation of algebraic integers and polynomial iteration”, in Sergiy Kolyada, Yuri Nanin, Martin Möller, Pieter Moree, Thomas Ward, editors, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, American Mathematical Society, page 83",
"text": "We consider the problem of finding all algebraic integers which belong to a bounded subset of the complex plane together with their conjugates.",
"type": "quotation"
}
],
"glosses": [
"A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients."
],
"holonyms": [
{
"word": "ring of integers"
}
],
"hypernyms": [
{
"word": "algebraic number"
}
],
"hyponyms": [
{
"word": "cyclotomic integer"
},
{
"word": "phi"
},
{
"word": "golden ratio"
},
{
"word": "root of unity"
},
{
"sense": "Gaussian integer, Eisenstein integer",
"word": "integer"
},
{
"sense": "Gaussian integer, Eisenstein integer",
"word": "rational integer"
}
],
"id": "en-algebraic_integer-en-noun-Rgog6KwS",
"links": [
[
"algebra",
"algebra"
],
[
"number theory",
"number theory"
],
[
"number field",
"number field"
],
[
"root",
"root"
],
[
"monic",
"monic"
],
[
"polynomial",
"polynomial"
],
[
"coefficient",
"coefficient"
],
[
"integer",
"integer"
],
[
"algebraic number",
"algebraic number"
],
[
"minimal polynomial",
"minimal polynomial"
]
],
"raw_glosses": [
"(algebra, number theory) A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients."
],
"related": [
{
"word": "quadratic integer"
}
],
"topics": [
"algebra",
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "代數整數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "dàishù zhěngshù",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "代数整数"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "algebrallinen kokonaisluku"
},
{
"code": "fr",
"lang": "French",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "entier algébrique"
},
{
"code": "el",
"lang": "Greek",
"roman": "algevrikós akéraios",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "αλγεβρικός ακέραιος"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "algebrai egész"
},
{
"code": "it",
"lang": "Italian",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "intero algebrico"
},
{
"code": "es",
"lang": "Spanish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "número entero algebraico"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"neuter"
],
"word": "algebraiskt heltal"
}
]
}
],
"word": "algebraic integer"
}
{
"forms": [
{
"form": "algebraic integers",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "algebraic integer (plural algebraic integers)",
"name": "en-noun"
}
],
"holonyms": [
{
"word": "ring of integers"
}
],
"hypernyms": [
{
"word": "algebraic number"
}
],
"hyponyms": [
{
"word": "cyclotomic integer"
},
{
"word": "phi"
},
{
"word": "golden ratio"
},
{
"sense": "Gaussian integer, Eisenstein integer",
"word": "integer"
},
{
"sense": "Gaussian integer, Eisenstein integer",
"word": "rational integer"
},
{
"word": "root of unity"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "quadratic 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 redundant transliterations",
"en:Algebra",
"en:Number theory",
"en:Numbers"
],
"examples": [
{
"text": "A Gaussian integer z=a+ib is an algebraic integer since it is a solution of either the equation z²+(-2a)z+(a²+b²)=0 or the equation z-a=0."
},
{
"ref": "1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 62",
"text": "An algebraic number is said to be an algebraic integer if the coefficient of the highest power of x in the minimal polynomial P is 1. The algebraic integers in an algebraic number field k form a ring R.",
"type": "quotation"
},
{
"text": "1989, Heinrich Rolletschek, Shortest Division Chains in Imaginary Quadratic Number Fields, Patrizia Gianni (editor), Symbolic and Algebraic Computation: International Symposium, Springer, LNCS 358, page 231,\nLet O_d be the set of algebraic integers in an imaginary quadratic number field Q [√],d<0, where d is the discriminant of O_d."
},
{
"ref": "2010, Pierre Moussa, “Localisation of algebraic integers and polynomial iteration”, in Sergiy Kolyada, Yuri Nanin, Martin Möller, Pieter Moree, Thomas Ward, editors, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, American Mathematical Society, page 83",
"text": "We consider the problem of finding all algebraic integers which belong to a bounded subset of the complex plane together with their conjugates.",
"type": "quotation"
}
],
"glosses": [
"A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients."
],
"links": [
[
"algebra",
"algebra"
],
[
"number theory",
"number theory"
],
[
"number field",
"number field"
],
[
"root",
"root"
],
[
"monic",
"monic"
],
[
"polynomial",
"polynomial"
],
[
"coefficient",
"coefficient"
],
[
"integer",
"integer"
],
[
"algebraic number",
"algebraic number"
],
[
"minimal polynomial",
"minimal polynomial"
]
],
"raw_glosses": [
"(algebra, number theory) A real or complex number (more generally, an element of a number field) which is a root of a monic polynomial whose coefficients are integers; equivalently, an algebraic number whose minimal polynomial (lowest-degree polynomial of which it is a root and whose leading coefficient is 1) has integer coefficients."
],
"topics": [
"algebra",
"mathematics",
"number-theory",
"sciences"
]
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "代數整數"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "dàishù zhěngshù",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "代数整数"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "algebrallinen kokonaisluku"
},
{
"code": "fr",
"lang": "French",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "entier algébrique"
},
{
"code": "el",
"lang": "Greek",
"roman": "algevrikós akéraios",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "αλγεβρικός ακέραιος"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"word": "algebrai egész"
},
{
"code": "it",
"lang": "Italian",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "intero algebrico"
},
{
"code": "es",
"lang": "Spanish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"masculine"
],
"word": "número entero algebraico"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "algebraic number whose minimal polynomial has integer coefficients",
"tags": [
"neuter"
],
"word": "algebraiskt heltal"
}
],
"word": "algebraic integer"
}
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.