See root of unity in All languages combined, or Wiktionary
Download JSON data for root of unity meaning in English (3.4kB)
{
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"form": "roots of unity",
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"lang_code": "en",
"pos": "noun",
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"examples": [
{
"text": "In the case of the field of complex numbers, it follows from de Moivre's formula that the n nth roots of unity are #x5C;textstyle#x5C;cos#x5C;left(k#x5C;frac#x7B;2#x5C;pi#x7D;#x7B;n#x7D;#x5C;right)#x2B;i#x5C;sin#x5C;left(k#x5C;frac#x7B;2#x5C;pi#x7D;#x7B;n#x7D;#x5C;right), where k#x3D;1,#x5C;dots,n.",
"type": "example"
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{
"ref": "2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89",
"text": "We now show that the primitive n-th roots of unity generate the other n-th roots of unity.",
"type": "quotation"
},
{
"ref": "2003, Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, page 72",
"text": "A nice application of Hensel's Lemma is to determine which roots of unity can be found in #x5C;Q#x5F;p.",
"type": "quotation"
},
{
"ref": "2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150",
"text": "We have seen that, for a fixed value of n, the multiplicative group (U#x5F;n,#x5C;dot) is generated by any primitive nth root of unity. In particular, if #x5C;omega is a primitive 6th root of unity, then #x5C;omega⁶#x3D;1, six is the smallest positive integer for which this is true, and U#x5F;6#x3D;#x5C;#x7B;#x5C;omega⁰,#x5C;omega,#x5C;omega²,#x5C;omega³,#x5C;omega⁴,#x5C;omega⁵#x5C;#x7D;. It is easy to see that #x5C;omega², which is a 6th root of unity, is also a cube root of unity. The same is true of #x5C;omega⁴. The element #x5C;omega³ is a square root of unity, whereas #x5C;omega⁵ is primitive.",
"type": "quotation"
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],
"glosses": [
"An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1."
],
"holonyms": [
{
"word": "circle group"
},
{
"word": "Kummer ring"
}
],
"hypernyms": [
{
"word": "algebraic integer"
}
],
"id": "en-root_of_unity-en-noun-4aHBERl-",
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"raw_glosses": [
"(number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1."
],
"topics": [
"mathematics",
"number-theory",
"sciences"
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "racine de l’unité"
},
{
"code": "de",
"lang": "German",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "Einheitswurzel"
},
{
"code": "it",
"lang": "Italian",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "radice dell'unità"
},
{
"code": "es",
"lang": "Spanish",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "raíz de la unidad"
}
],
"wikipedia": [
"root of unity"
]
}
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"word": "root of unity"
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"examples": [
{
"text": "In the case of the field of complex numbers, it follows from de Moivre's formula that the n nth roots of unity are #x5C;textstyle#x5C;cos#x5C;left(k#x5C;frac#x7B;2#x5C;pi#x7D;#x7B;n#x7D;#x5C;right)#x2B;i#x5C;sin#x5C;left(k#x5C;frac#x7B;2#x5C;pi#x7D;#x7B;n#x7D;#x5C;right), where k#x3D;1,#x5C;dots,n.",
"type": "example"
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{
"ref": "2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89",
"text": "We now show that the primitive n-th roots of unity generate the other n-th roots of unity.",
"type": "quotation"
},
{
"ref": "2003, Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, page 72",
"text": "A nice application of Hensel's Lemma is to determine which roots of unity can be found in #x5C;Q#x5F;p.",
"type": "quotation"
},
{
"ref": "2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150",
"text": "We have seen that, for a fixed value of n, the multiplicative group (U#x5F;n,#x5C;dot) is generated by any primitive nth root of unity. In particular, if #x5C;omega is a primitive 6th root of unity, then #x5C;omega⁶#x3D;1, six is the smallest positive integer for which this is true, and U#x5F;6#x3D;#x5C;#x7B;#x5C;omega⁰,#x5C;omega,#x5C;omega²,#x5C;omega³,#x5C;omega⁴,#x5C;omega⁵#x5C;#x7D;. It is easy to see that #x5C;omega², which is a 6th root of unity, is also a cube root of unity. The same is true of #x5C;omega⁴. The element #x5C;omega³ is a square root of unity, whereas #x5C;omega⁵ is primitive.",
"type": "quotation"
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"glosses": [
"An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1."
],
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"(number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1."
],
"topics": [
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"number-theory",
"sciences"
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"translations": [
{
"code": "fr",
"lang": "French",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "racine de l’unité"
},
{
"code": "de",
"lang": "German",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "Einheitswurzel"
},
{
"code": "it",
"lang": "Italian",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "radice dell'unità"
},
{
"code": "es",
"lang": "Spanish",
"sense": "field element, some positive power of which equals 1",
"tags": [
"feminine"
],
"word": "raíz de la unidad"
}
],
"word": "root of unity"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 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.
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