Essential extension - Revision history

66.74.176.59: edit

2015-01-05T16:06:10Z

edit

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	~~In [[abstract algebra]]~~, a '''rupture field''' of a [[polynomial]] <math>P(X)</math> over a given [[field (mathematics)\|field]] <math>K</math> such that <math>P(X)\in K[X]</math> is the [[field extension]] of <math>K</math> generated by a [[root of a function\|root]] <math>a</math> of <math>P(X)</math>.~~<ref>{{Cite book~~		Bricklayer Darell from Rockland, likes to spend some time acting, como ganhar dinheiro na internet and car. Enjoys travel and ended up encouraged after visiting Notre-Dame Cathedral in Tournai.<br><br>Look into my blog post [http://ganhandodinheironainternet.comoganhardinheiro101.com/ como conseguir dinheiro]
	~~\| last = Escofier~~
	~~\| first = Jean-Paul~~
	~~\| title = Galois Theory~~
	~~\| publisher = Springer~~
	~~\| date = 2001~~
	~~\| pages = 62~~
	~~\| isbn = 0-387-98765-7}}~~
	~~</ref>~~

	~~For instance, if <math>K=\mathbb Q</math>~~ and ~~<math>P(X)=X^3~~-~~2</math> then <math>\mathbb Q[\sqrt[3]2]</math> is a rupture field for <math>P(X)</math>.~~

	The notion is interesting mainly if <math>P(X)</math> is [[irreducible polynomial\|irreducible]] over <math>K</math>. In that case, all rupture fields of <math>P(X)</math> over <math>K</math> are isomorphic, non canonically, to <math>K_P=K[X]/(P(X))</math>: if <math>L=K[a]</math> where <math>a</math> is a root of <math>P(X)</math>, then the [[ring homomorphism]] <math>f</math> defined by <math>f(k)=k</math> for all <math>k\in ~~K</math> and <math>f(X\mod P)=a</math> is an [[isomorphism]]~~. ~~Also, in this case the degree of the extension equals the degree of~~ <~~math~~>P<~~/math~~>.

	~~The rupture field of a [[polynomial]] does not necessarily contain all the roots of that [~~[~~polynomial]]~~: ~~in the above example the field <math>\mathbb Q[\sqrt[3]2]<~~/~~math> does not contain the other two (complex) roots of <math>P(X)<~~/~~math> (namely <math>\omega\sqrt[3]2</math> and <math>\omega^2\sqrt[3]2</math> where <math>\omega</math> is a primitive third root of unity)~~. ~~For a field containing all the roots of a [[polynomial]], see the [[splitting field]]~~.

	~~==Examples==~~
	~~The rupture field of <math>X^2+1<~~/~~math> over <math>\mathbb R</math> is <math>\mathbb C</math>. It is also its [[splitting field]].~~

	The rupture field of <math>X^2+1</math> over <math>\mathbb F_3</math> is <math>\mathbb F_9</math> since there is no element of <math>\mathbb F_3</math> with square equal to <math>-1</math> (and all quadratic [[field extension\|extensions]] of <math>\mathbb F_3</math> are isomorphic to <math>\mathbb F_9</math>).

	~~==See also==~~
	* [[Splitting field]]
	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Rupture Field}}~~
	~~[[Category:Field theory]~~]

130.235.3.80: /* Properties */

2013-11-14T12:56:54Z

Properties

← Older revision		Revision as of 14:56, 14 November 2013
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	~~Greetings! I am Myrtle Shroyer. Since she was eighteen she~~'~~s been operating as~~ a ~~receptionist but her promotion never comes. It's not~~ a ~~common factor but what she likes performing~~ is ~~base leaping~~ and ~~now she~~ is ~~trying to earn cash with it~~. ~~South Dakota~~ is ~~exactly~~ where ~~I've always been residing~~.<br><br>~~Review my webpage :~~: [~~http:~~//~~raybana~~.~~com~~/~~chat~~/pg/~~profile~~/~~GIsaachse raybana~~.~~com~~]		In [[abstract algebra]], a '''rupture field''' of a [[polynomial]] <math>P(X)</math> over a given [[field (mathematics)\|field]] <math>K</math> such that <math>P(X)\in K[X]</math> is the [[field extension]] of <math>K</math> generated by a [[root of a function\|root]] <math>a</math> of <math>P(X)</math>.<ref>{{Cite book
			\| last = Escofier
			\| first = Jean-Paul
			\| title = Galois Theory
			\| publisher = Springer
			\| date = 2001
			\| pages = 62
			\| isbn = 0-387-98765-7}}
			</ref>

			For instance, if <math>K=\mathbb Q</math> and <math>P(X)=X^3-2</math> then <math>\mathbb Q[\sqrt[3]2]</math> is a rupture field for <math>P(X)</math>.

			The notion is interesting mainly if <math>P(X)</math> is [[irreducible polynomial\|irreducible]] over <math>K</math>. In that case, all rupture fields of <math>P(X)</math> over <math>K</math> are isomorphic, non canonically, to <math>K_P=K[X]/(P(X))</math>: if <math>L=K[a]</math> where <math>a</math> is a root of <math>P(X)</math>, then the [[ring homomorphism]] <math>f</math> defined by <math>f(k)=k</math> for all <math>k\in K</math> and <math>f(X\mod P)=a</math> is an [[isomorphism]]. Also, in this case the degree of the extension equals the degree of <math>P</math>.

			The rupture field of a [[polynomial]] does not necessarily contain all the roots of that [[polynomial]]: in the above example the field <math>\mathbb Q[\sqrt[3]2]</math> does not contain the other two (complex) roots of <math>P(X)</math> (namely <math>\omega\sqrt[3]2</math> and <math>\omega^2\sqrt[3]2</math> where <math>\omega</math> is a primitive third root of unity). For a field containing all the roots of a [[polynomial]], see the [[splitting field]].

			==Examples==
			The rupture field of <math>X^2+1</math> over <math>\mathbb R</math> is <math>\mathbb C</math>. It is also its [[splitting field]].

			The rupture field of <math>X^2+1</math> over <math>\mathbb F_3</math> is <math>\mathbb F_9</math> since there is no element of <math>\mathbb F_3</math> with square equal to <math>-1</math> (and all quadratic [[field extension\|extensions]] of <math>\mathbb F_3</math> are isomorphic to <math>\mathbb F_9</math>).

			==See also==
			* [[Splitting field]]
			==References==
			{{reflist}}

			{{DEFAULTSORT:Rupture Field}}
			[[Category:Field theory]]

en>Rschwieb: clarification that it is not really a "new" definition

2012-06-29T16:18:39Z

clarification that it is not really a "new" definition

New page

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