See Galois extension in All languages combined, or Wiktionary
Download JSON data for Galois extension meaning in English (4.2kB)
{
"etymology_text": "Named for its connection with Galois theory and after French mathematician Évariste Galois.",
"forms": [
{
"form": "Galois extensions",
"tags": [
"plural"
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"expansion": "Galois extension (plural Galois extensions)",
"name": "en-noun"
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"langcode": "en",
"name": "Algebra",
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"derived": [
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"word": "differential Galois extension"
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],
"examples": [
{
"text": "The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.",
"type": "example"
},
{
"text": "The fundamental theorem of Galois theory states that there is a one-to-one correspondence between the subfields of a Galois extension and the subgroups of its Galois group.",
"type": "example"
},
{
"ref": "1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108",
"text": "Corollary If L#x3A;K is a Galois extension, there exists an irreducible polynomial f in K#x5B;x#x5D; such that L#x3A;K is a splitting field extension for f over K.",
"type": "quotation"
},
{
"ref": "1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642",
"text": "First, arithmetic obstructions against constructing central extensions of a fixed finite base Galois extension are analyzed with the local-global principle to give some quantitative estimates of them.",
"type": "quotation"
},
{
"ref": "2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211",
"text": "With the help of the results in Section 7.5 it is not hard to describe all Galois extensions.\nProposition 7.6.1. Let E#x2F;F be a finite field extension. Then (i) E#x2F;F is a Galois extension if and only if it is normal and separable; (ii) E#x2F;F is contained in a Galois extension if and only if it is separable.",
"type": "quotation"
}
],
"glosses": [
"An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F."
],
"hypernyms": [
{
"word": "algebraic extension"
},
{
"word": "normal extension"
},
{
"word": "separable extension"
}
],
"id": "en-Galois_extension-en-noun-vgqlOQJp",
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"algebra",
"algebra"
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"algebraic extension",
"algebraic extension"
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"normal",
"normal extension"
],
[
"separable extension",
"separable extension"
],
[
"fixed field",
"fixed field"
],
[
"automorphism group",
"automorphism group"
],
[
"Galois group",
"Galois group"
]
],
"qualifier": "Galois theory",
"raw_glosses": [
"(algebra, Galois theory) An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F."
],
"related": [
{
"word": "Galois group"
},
{
"word": "Hopf-Galois extension"
}
],
"topics": [
"algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "extension de Galois"
},
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "extension galoisienne"
},
{
"code": "it",
"lang": "Italian",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "estensione di Galois"
}
],
"wikipedia": [
"Galois extension",
"Évariste Galois"
]
}
],
"word": "Galois extension"
}
{
"derived": [
{
"word": "differential Galois extension"
}
],
"etymology_text": "Named for its connection with Galois theory and after French mathematician Évariste Galois.",
"forms": [
{
"form": "Galois extensions",
"tags": [
"plural"
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}
],
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"expansion": "Galois extension (plural Galois extensions)",
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{
"word": "algebraic extension"
},
{
"word": "normal extension"
},
{
"word": "separable extension"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "Galois group"
},
{
"word": "Hopf-Galois extension"
}
],
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"examples": [
{
"text": "The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.",
"type": "example"
},
{
"text": "The fundamental theorem of Galois theory states that there is a one-to-one correspondence between the subfields of a Galois extension and the subgroups of its Galois group.",
"type": "example"
},
{
"ref": "1986, D. J. H. Garling, A Course in Galois Theory, Cambridge University Press, page 108",
"text": "Corollary If L#x3A;K is a Galois extension, there exists an irreducible polynomial f in K#x5B;x#x5D; such that L#x3A;K is a splitting field extension for f over K.",
"type": "quotation"
},
{
"ref": "1989, Katsuya Miyake, “On central extensions”, in Jean-Marie De Koninck, Claude Levesque, editors, Number Theory, Walter de Gruyter, page 642",
"text": "First, arithmetic obstructions against constructing central extensions of a fixed finite base Galois extension are analyzed with the local-global principle to give some quantitative estimates of them.",
"type": "quotation"
},
{
"ref": "2003, Paul M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, page 211",
"text": "With the help of the results in Section 7.5 it is not hard to describe all Galois extensions.\nProposition 7.6.1. Let E#x2F;F be a finite field extension. Then (i) E#x2F;F is a Galois extension if and only if it is normal and separable; (ii) E#x2F;F is contained in a Galois extension if and only if it is separable.",
"type": "quotation"
}
],
"glosses": [
"An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F."
],
"links": [
[
"algebra",
"algebra"
],
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"algebraic extension"
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[
"normal",
"normal extension"
],
[
"separable extension",
"separable extension"
],
[
"fixed field",
"fixed field"
],
[
"automorphism group",
"automorphism group"
],
[
"Galois group",
"Galois group"
]
],
"qualifier": "Galois theory",
"raw_glosses": [
"(algebra, Galois theory) An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F."
],
"topics": [
"algebra",
"mathematics",
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"wikipedia": [
"Galois extension",
"Évariste Galois"
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"translations": [
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "extension de Galois"
},
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "extension galoisienne"
},
{
"code": "it",
"lang": "Italian",
"sense": "algebraic extension that is normal and separable",
"tags": [
"feminine"
],
"word": "estensione di Galois"
}
],
"word": "Galois extension"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-03 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.
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