See Galois extension on Wiktionary
{
"etymology_text": "Named for its connection with Galois theory and after French mathematician Évariste Galois.",
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"derived": [
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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": "quote"
},
{
"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": "quote"
},
{
"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.",
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"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."
],
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"word": "algebraic extension"
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"word": "normal extension"
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"word": "separable extension"
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"id": "en-Galois_extension-en-noun-vgqlOQJp",
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"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"
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{
"word": "Hopf-Galois extension"
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"translations": [
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
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"word": "extension de Galois"
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{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
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"word": "extension galoisienne"
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{
"code": "it",
"lang": "Italian",
"sense": "algebraic extension that is normal and separable",
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"word": "estensione di Galois"
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"wikipedia": [
"Galois extension",
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"word": "Galois extension"
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"etymology_text": "Named for its connection with Galois theory and after French mathematician Évariste Galois.",
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"word": "normal extension"
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"word": "Hopf-Galois extension"
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"text": "The significance of a Galois extension is that it has a Galois group and obeys the fundamental theorem of Galois theory.",
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{
"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.",
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},
{
"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": "quote"
},
{
"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": "quote"
},
{
"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.",
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],
"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."
],
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"qualifier": "Galois theory",
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"(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": [
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"translations": [
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
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"feminine"
],
"word": "extension de Galois"
},
{
"code": "fr",
"lang": "French",
"sense": "algebraic extension that is normal and separable",
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"word": "extension galoisienne"
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{
"code": "it",
"lang": "Italian",
"sense": "algebraic extension that is normal and separable",
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"word": "estensione di Galois"
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"word": "Galois extension"
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