Partition matroid - Revision history

en>David Eppstein: /* Clique complexes */ better wlink

2014-04-22T00:43:17Z

Clique complexes: better wlink

← Older revision		Revision as of 02:43, 22 April 2014
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	In ~~mathematics~~, a ~~'''quadratically closed field'''~~ is ~~a [[field (mathematics)\|field]] in which every element~~ of the ~~field~~ has a ~~[[square root]] in the field~~.<~~ref name=Lam33~~>~~Lam (2005) p~~.~~ 33<~~/~~ref><ref name~~=~~R230>Rajwade (1993) p. 230~~</~~ref~~>		In Milan this unique Sept, the errant indicate by means of Prada must have been a actual News.<br>This helmet is made of super strong ABS Thermoplastic Shell that offers you the protection and durability that you need. It also has D ring fasteners for a more secure enclosure and is manufactured to meet and even exceed DOT standards for motorcycle helmets.<br>http://www.bendtrapclub.com/cheap/ugg.asp?p=358 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=430 <br /> http://www.bendtrapclub.com/cheap/ugg.asp? When you loved this article and you would love to receive more info with regards to [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] assure visit our site. p=46 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=223 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=403 <br /> <br>http://atlantaka.org/?document_srl=926201<br>http://www.tinterovirtual.com/modules.php?name=Your_Account&op=userinfo&username=LWhitacre
	~~==Examples==~~
	* The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]] is quadratically closed.
	* The field of real numbers is not quadratically closed as it does not contain a square root of −1.
	* The union of the [[finite field]]s <math>F_{5^{2^n}}</~~math> for ''n'' ≥ 0 is quadratically closed but not algebraically closed~~.<~~ref name=Lam34~~/>
	* The field of [[constructible number]]s is quadratically closed but not algebraically closed.~~<ref name=Lam220>Lam (2005) p~~.~~ 220<~~/~~ref>~~

	~~==Properties==~~
	* A field is quadratically closed if and ~~only if it has~~ [~~[universal invariant]] equal to 1~~.
	* Every quadratically closed field is a [[Pythagorean field]] but not conversely (for example, '''R''' is Pythagorean); however, every non-[[formally real]] Pythagorean field is quadratically closed.~~<ref name=R230~~/>
	* A field is quadratically closed if and only if its [[Witt–Grothendieck ring]~~] is isomorphic to '''Z''' under the dimension mapping~~.~~<ref name~~=~~Lam34>Lam (2005) p. 34~~</~~ref~~>
	* A formally real [[Euclidean field]] ''E'' is not quadratically closed (as −1 is not a square in ''E'') but the quadratic extension ''E''(√−1) is quadratically closed.<ref name=Lam220/>
	* Let ''E''/~~''F'' be a finite [[field extension\|extension]] where ''E'' is quadratically closed~~. ~~Either −1 is a square in ''F'' and ''F'' is quadratically closed, or −1 is not a square in ''F'' and ''F'' is Euclidean~~. ~~This "going-down theorem" may be deduced from the [[Diller–Dress theorem]]~~.~~<ref name~~=~~Lam270>Lam (2005) p.270~~</~~ref~~>

	~~==Quadratic closure==~~
	~~A '''quadratic closure''' of a field ''F'' is a quadratically closed field which embeds in any other quadratically closed field containing ''F''.~~ ~~A quadratic closure for any given ''F'' may be constructed as a subfield of the [[algebraic closure]] ''F''<sup>alg<~~/~~sup> of ''F'', as the union of all quadratic extensions of ''F'' in ''F''<sup>alg<~~/~~sup>~~.~~<ref name=Lam220~~/>

	~~===Examples=~~==
	* The quadratic closure of '''R''' is '''C'''.<~~ref name=Lam220~~/>
	* The quadratic closure of '''F'''<~~sub~~>5</~~sub> is the union of the~~ <~~math~~>~~F_{5^{2^n}}<~~/~~math>.<ref name=Lam220~~/>
	* The quadratic closure of '''Q''' is the field of constructible numbers.

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.~~11023 \| mr = 2104929 }}~~
	* {{cite book \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. ~~R. \| last~~=~~Rajwade \| publisher~~=~~[[Cambridge University Press]] \| year~~=~~1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}~~

	~~[[Category:Field theory]]~~

en>David Eppstein: /* Definition */ link Matroid rank

2012-09-26T06:07:36Z

Definition: link Matroid rank

← Older revision		Revision as of 08:07, 26 September 2012
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	~~Take~~ a ~~look at Ford~~'~~s seminal Model T it looks more like~~ a ~~horse If you loved this article and you would like to get more details pertaining to~~ [~~http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots~~] ~~kindly take~~ a ~~look at~~ the ~~internet site~~. ~~carriage minus the horses a very boxy design, indeed~~.<br>~~Shes my only real friend~~, ~~besides this one kid, who Im pretty sure~~ is ~~only hanging around me because he~~ is ~~mentally challenged~~. ~~I guess hes~~ the ~~only one that can tolerate me.~~<br>~~http:~~//~~www~~.~~hamiltonorchards~~.~~com~~/~~kors/?p~~=~~144~~ <br /> ~~http://www~~.~~hamiltonorchards~~.~~com/kors/?p=107~~ <br /> ~~http://www~~.~~hamiltonorchards.com/kors/?p~~=~~106 <br~~ /> ~~http://www~~.~~hamiltonorchards~~.~~com/kors/?~~p~~=177~~ <br /> ~~http:~~//~~www~~.~~hamiltonorchards.com~~/~~kors/?p~~=~~203~~ <br /> <br>~~http:~~//~~www~~.~~darkasylum.org/wiki~~/~~Sick_And_Tired_Of_Doing_Bags_Outlet_The_Old_Way_Read_This<br~~>~~http://gracechapel~~.~~kr/?document_srl~~=~~435423~~		In mathematics, a '''quadratically closed field''' is a [[field (mathematics)\|field]] in which every element of the field has a [[square root]] in the field.<ref name=Lam33>Lam (2005) p. 33</ref><ref name=R230>Rajwade (1993) p. 230</ref>
			==Examples==
			* The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]] is quadratically closed.
			* The field of real numbers is not quadratically closed as it does not contain a square root of −1.
			* The union of the [[finite field]]s <math>F_{5^{2^n}}</math> for ''n'' ≥ 0 is quadratically closed but not algebraically closed.<ref name=Lam34/>
			* The field of [[constructible number]]s is quadratically closed but not algebraically closed.<ref name=Lam220>Lam (2005) p. 220</ref>

			==Properties==
			* A field is quadratically closed if and only if it has [[universal invariant]] equal to 1.
			* Every quadratically closed field is a [[Pythagorean field]] but not conversely (for example, '''R''' is Pythagorean); however, every non-[[formally real]] Pythagorean field is quadratically closed.<ref name=R230/>
			* A field is quadratically closed if and only if its [[Witt–Grothendieck ring]] is isomorphic to '''Z''' under the dimension mapping.<ref name=Lam34>Lam (2005) p. 34</ref>
			* A formally real [[Euclidean field]] ''E'' is not quadratically closed (as −1 is not a square in ''E'') but the quadratic extension ''E''(√−1) is quadratically closed.<ref name=Lam220/>
			* Let ''E''/''F'' be a finite [[field extension\|extension]] where ''E'' is quadratically closed. Either −1 is a square in ''F'' and ''F'' is quadratically closed, or −1 is not a square in ''F'' and ''F'' is Euclidean. This "going-down theorem" may be deduced from the [[Diller–Dress theorem]].<ref name=Lam270>Lam (2005) p.270</ref>

			==Quadratic closure==
			A '''quadratic closure''' of a field ''F'' is a quadratically closed field which embeds in any other quadratically closed field containing ''F''. A quadratic closure for any given ''F'' may be constructed as a subfield of the [[algebraic closure]] ''F''<sup>alg</sup> of ''F'', as the union of all quadratic extensions of ''F'' in ''F''<sup>alg</sup>.<ref name=Lam220/>

			===Examples===
			* The quadratic closure of '''R''' is '''C'''.<ref name=Lam220/>
			* The quadratic closure of '''F'''<sub>5</sub> is the union of the <math>F_{5^{2^n}}</math>.<ref name=Lam220/>
			* The quadratic closure of '''Q''' is the field of constructible numbers.

			==References==
			{{reflist}}
			* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}
			* {{cite book \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

			[[Category:Field theory]]

en>David Eppstein: /* Properties */ link dual matroid

2012-08-29T00:57:46Z

Properties: link dual matroid

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