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- In [[mathematics]], a '''field extension''' of a [[field (mathematics)|field]] ''F'' is a field ''E'' such that ''F' ...'E'' is a [[vector space]] over ''F''. The ''degree'' or ''index'' of the field extension [''E'':''F''] is the [[dimension]] of ''E'' as an ''F''-vector space. The3 KB (435 words) - 22:38, 22 February 2009
- 83 bytes (12 words) - 13:18, 21 December 2008
- Auto-populated based on [[Special:WhatLinksHere/Field extension]]. Needs checking by a human.857 bytes (112 words) - 16:32, 11 January 2010
Page text matches
- ...c number theory]], the '''conductor''' or '''relative conductor''' of an [[field extension|extension]] of [[algebraic number field]]s is a [[modulus (algebraic number1 KB (177 words) - 01:07, 18 February 2009
- #REDIRECT [[Field extension#Simple extension]]46 bytes (5 words) - 02:18, 20 December 2008
- #REDIRECT [[Field extension#Simple extension]]46 bytes (5 words) - 02:18, 20 December 2008
- #REDIRECT [[Field extension#Separable extension]]49 bytes (5 words) - 02:26, 20 December 2008
- #REDIRECT [[Field extension#Algebraic extension]]49 bytes (5 words) - 11:53, 22 December 2008
- #REDIRECT [[Field extension#Separable extension]]49 bytes (5 words) - 02:27, 20 December 2008
- #REDIRECT [[Field extension#Separable extension]]49 bytes (5 words) - 02:30, 20 December 2008
- ...gebra]], a '''normal extension''' of [[field (mathematics)|fields]] is a [[field extension]] ''E''/''F'' which contains all the roots of an irreducible polynomial if568 bytes (88 words) - 17:21, 7 February 2009
- In [[mathematics]], a '''field extension''' of a [[field (mathematics)|field]] ''F'' is a field ''E'' such that ''F' ...'E'' is a [[vector space]] over ''F''. The ''degree'' or ''index'' of the field extension [''E'':''F''] is the [[dimension]] of ''E'' as an ''F''-vector space. The3 KB (435 words) - 22:38, 22 February 2009
- A field extension generated by the roots of all polynomials over the ground field.118 bytes (17 words) - 17:06, 6 January 2009
- Auto-populated based on [[Special:WhatLinksHere/Field extension]]. Needs checking by a human.857 bytes (112 words) - 16:32, 11 January 2010
- #REDIRECT [[Field extension]]29 bytes (3 words) - 04:13, 20 December 2008
- {{r|Field extension}}530 bytes (68 words) - 19:04, 11 January 2010
- A field extension in which all elements are separable.90 bytes (12 words) - 04:12, 20 December 2008
- {{r|Field extension}}1 KB (146 words) - 16:32, 11 January 2010
- An element of a field extension for which the minimal polynomial has distinct roots.120 bytes (17 words) - 04:11, 20 December 2008
- A field extension which contains all the roots of an irreducible polynomial if it contains on141 bytes (21 words) - 13:16, 21 December 2008
- {{r|Field extension}}644 bytes (86 words) - 19:50, 11 January 2010
- A field extension generated by the roots of a polynomial.93 bytes (13 words) - 13:13, 21 December 2008
- ...-Schreier polynomial''' is a polynomial whose roots are used to generate [[field extension]]s of [[prime number|prime]] degree ''p'' in [[characteristic of a field|ch2 KB (295 words) - 15:43, 7 December 2008
- {{r|Field extension}}692 bytes (91 words) - 16:33, 11 January 2010
- {{r|Field extension}}584 bytes (79 words) - 15:48, 11 January 2010
- ...criminant of an algebraic number field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric st1 KB (235 words) - 01:20, 18 February 2009
- A field extension of the rational numbers of finite degree; a principal object of study in al151 bytes (22 words) - 03:01, 1 January 2009
- {{r|Field extension}}595 bytes (77 words) - 15:38, 11 January 2010
- {{r|Field extension}}554 bytes (72 words) - 16:00, 11 January 2010
- ...s the [[ramification#In algebraic number theory|ramification]] data of the field extension ''L''/''K''. A prime ideal ''p'' of ''K'' ramifies in ''L'' if and only if2 KB (382 words) - 09:40, 12 June 2009
- {{r|Field extension}}1 KB (169 words) - 08:53, 22 December 2008
- ...bgroup <math>Aut_L(K)</math> of the full automorphism group of ''K''. A [[field extension]] <math>K/L</math> of finite index ''d'' is ''[[normal extension|normal]]''3 KB (418 words) - 12:18, 20 December 2008
- {{r|Field extension}}544 bytes (70 words) - 18:34, 11 January 2010
- ..., a '''splitting field''' for a polynomial ''f'' over a field ''F'' is a [[field extension]] ''E''/''F'' with the properties that ''f'' splits completely over ''E'',1 KB (147 words) - 09:16, 4 July 2009
- ...''n''-th root of unity, then the ''n''-th cyclotomic field ''F'' is the [[field extension]] <math>\mathbf{Q}(\zeta)</math>.2 KB (342 words) - 12:52, 21 January 2009
- ...quadratic field''' is a [[Field theory (mathematics)|field]] which is an [[field extension|extension]] of its [[prime field]] of degree two.3 KB (453 words) - 17:18, 6 February 2009
- {{r|Field extension}}710 bytes (90 words) - 19:54, 11 January 2010
- {{r|Field extension}}909 bytes (144 words) - 13:19, 21 December 2008
- ...The degree of the minimal polynomial of α is equal to the degree of the [[field extension]] '''Q'''(α)/'''Q'''. ...tisfies. The polynonmial ring ''F''[α] is then a field, and is the simple field extension ''F''(α). This field is a finite dimension al vector space over ''F'', on4 KB (613 words) - 02:34, 4 January 2013
- Key concepts are [[Field extension|field extensions]] and [[Group theory|groups]], which should be thoroughly ...a subfield and also all the roots of ''f''. This field is known as the [[field extension|extension]] of ''K'' by the roots of ''f'', or the ''[[splitting field]]''4 KB (683 words) - 22:17, 7 February 2010
- {{r|Field extension}}2 KB (206 words) - 19:38, 11 January 2010
- {{r|Field extension}}551 bytes (71 words) - 18:22, 11 January 2010
- *[[Field extension]]3 KB (496 words) - 22:16, 7 February 2010
- In Δ is not a square in ''F'' then the [[field extension]] <math>F(\sqrt\Delta)</math> is [[quadratic field|quadratic]] over ''F'':10 KB (1,580 words) - 08:52, 4 March 2009
- Let us take the case that ''G'' is the [[Galois group]] of a [[field extension]] ''L''/''K''. A factor system in H<sup>2</sup>(''G'',''L''<sup>*</sup>) g3 KB (519 words) - 15:42, 2 January 2013
- An ''algebraic number field'' ''K'' is a finite degree [[field extension]] of the [[field (mathematics)|field]] '''Q''' of [[rational number]]s. Th7 KB (1,077 words) - 17:18, 10 January 2009
- ...e one. Since <math>\scriptstyle\mathbb{C} = \mathbb{R}[i]</math>, any such field extension also extends <math>\scriptstyle\mathbb{R}</math>. Now, any <math>\scriptsty5 KB (924 words) - 16:35, 11 December 2008
- * [[Algebraically independent set]]s in a [[field extension]];2 KB (334 words) - 16:29, 7 February 2009
- Let ''K'' be an [[algebraic number field]], a finite [[field extension|extension]] of '''Q''', and ''E'' an elliptic curve defined over ''K''. Th10 KB (1,637 words) - 16:03, 17 December 2008
- ...analysis, we could next show that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]18 KB (3,028 words) - 17:12, 25 August 2013
- ...analysis, we could next show that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]20 KB (3,304 words) - 17:11, 25 August 2013