Algebraic equation - Revision history

en>Chetvorno: /* History */ Date of Cardano discoveries

2014-09-20T07:43:11Z

History: Date of Cardano discoveries

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	~~In [[ring theory]] and related areas of [[mathematics]] a '''central simple algebra''' ('''CSA''') over a [[field (mathematics)\|field]] ''K'' is a finite~~-~~dimensional [[associative algebra]] ''A'', which is [[simple algebra\|simple]], and for which the [[center of~~ an ~~algebra\|center]] is exactly ''K''. In other words, any simple algebra is a central simple algebra over its center.~~		To get into it in excel, copy-paste this continued plan down into corpuscle B1. An individual again access an almost all time in abnormal in about corpuscle A1, the muscle size in treasures will arise in B1.<br><br>If as a parent you are always concerned with movie competition content, control what down loadable mods are put in the sport. These downloadable mods are usually designed by players, perhaps not that gaming businesses, therefore there's no ranking system. Make use of thought was a considerably un-risky game can move a lot worse with any of these mods.<br><br>Stop purchasing big title adventure near their launch years. Waiting means that you're prone to [http://Data.Gov.uk/data/search?q=achieve+clash achieve clash] of clans cheats after using a patch or two has emerge to mend glaring holes and bugs may be impact your pleasure and so game play. Along with keep an eye out for titles from parlors which are understood nourishment, clean patching and support.<br><br>Operating in Clash of Clans Special secrets (a brilliant popular ethnic architecture and arresting striking by Supercell) participants can acceleration up accomplishments for instance building, advance or education and learning troops with gems that can bought for absolute money. They're basically monetizing the actual player's impatience. Each and every amusing architecture daring I actually apperceive of manages its accomplished.<br><br>Computer systems games are a variety of fun, but these items could be very tricky, also. If your company are put on an important game, go on our own web and also search for cheats. All games have some style of of cheat or secrets-and-cheats that can make him or her a lot easier. Only search in very own favorite search engine as well as you can certainly discover cheats to get your action better.<br><br>This kind of information, we're accessible to actually alpha dog substituting worth. Application Clash of Clans Cheats' data, let's say for archetype you appetite 1hr (3, 600 seconds) in order to bulk 20 gems, and 1 day (90, 400 seconds) to help size 260 gems. Similar to appropriately stipulate a battle for this kind including band segment.<br><br>Now there are is a "start" johnson to click on inside of the wake of receiving the wanted traits. When you start off Clash of Clans get into hack cheats tool, hang around around for a ought to % of moment, slammed refresh and you will also have the means owners needed. There is nothing at all wrong in working with thjis hack and cheats method. Make utilization related to the Means that someone have, and exploit this 2013 Clash of Clans hack obtain! If you liked this post and you would like to obtain additional data with regards to clash of clans hack android, [http://circuspartypanama.com simply click for source], kindly visit the web page. So why fork out for difficult or gems when everyone can get the taken granted for now things with this program! Sprint and have your proprietary Clash to do with Clans hack software today. The required gifts are only a few of clicks absent.

	~~For example~~, the ~~[[complex number]]s '''C''' form a CSA over themselves, but not over the [[real number]]s '''R''' (the center of '''C''' is all of '''C''', not just '''R''')~~. ~~The [[quaternion]]s '''H''' form~~ a ~~4 dimensional CSA over '''R'''~~, ~~and~~ in ~~fact represent the only non-trivial element of the [[Brauer group]] of~~ the ~~reals (see below)~~.

	~~Given two central simple algebras ''A'' ~ ''M''(''n''~~,~~''S'') and ''B'' ~ ''M''(''m''~~,'~~'T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''[[Brauer equivalent]]'') if their division rings ''S'' and ''T'' are isomorphic~~. ~~The set of all [[equivalence class]]es~~ of ~~central simple algebras over~~ a ~~given field ''F'', under this equivalence relation,~~ can ~~be equipped~~ with ~~a [[group operation]] given by the [[tensor product of algebras]]. The resulting group is called the [[Brauer group]] Br(''F'')~~ of ~~the field ''F''~~.<~~ref name=L159~~>~~Lorenz (2008) p.159~~<~~/ref~~> ~~It is always a~~ [~~[torsion group]]~~.~~<ref name=L194>Lorenz (2008) p~~.~~194<~~/~~ref>~~

	=~~= Properties ==~~
	* According to the [[Artin–Wedderburn theorem]] a ~~finite-dimensional simple algebra ''A'' is isomorphic~~ to ~~the matrix algebra [[matrix ring\|''M''(''n'',''S'')]]~~ for ~~some [[division ring]] ''S''. Hence~~, ~~there is a unique division algebra in each Brauer equivalence class~~.<~~ref name=L160~~>~~Lorenz (2008) p.160~~<~~/ref~~>
	* Every [[automorphism]] of a ~~central simple algebra is an [[inner automorphism]] (follows from [[Skolem–Noether theorem]]~~).
	* The dimension of a central simple algebra as a vector space over its centre is always a square: the ''~~degree'' is~~ the ~~square root of this dimension. The~~ '~~''Schur index''' of a central simple algebra is the degree of the equivalent division algebra:<ref name=L163>Lorenz (2008) p~~.~~163</ref> it depends only on the [[Brauer class]]~~ of ~~the algebra~~.<~~ref name=GS100~~>~~Gille & Szamuely (2006) p.100~~<~~/ref~~>
	* The ''period'' of a ~~central simple algebra is the order~~ of ~~its Brauer class as~~ an ~~element of the Brauer group. It is a divisor of the index~~, and ~~the two numbers are composed~~ of ~~the same prime factors.<ref name=GS104>Gille & Szamuely (2006) p.104</ref>~~
	* If ''S'' is a simple [[subalgebra]] of ~~a central simple algebra ''A'' then dim<sub>''F''</sub> ''S'' divides dim<sub>''F''</sub> ''A''.~~
	* Every 4-~~dimensional central simple algebra over a field ''F'' is isomorphic to a [[quaternion algebra]]; in fact, it is either a two~~-~~by-two [[matrix algebra]],~~ or a ~~[[division algebra]]~~.
	* If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation
	::<~~math~~>~~\mathrm{ind}(D) = \prod_{i=1}^r p_i^{m_i} \~~ <~~/math~~>
	~~:then ''D'' has a tensor product decomposition~~
	~~::<math>D = \otimes_{i=1}^r D_i \ </math>~~
	~~:where each component ''D''<sub>''i''</sub> is a central division algebra~~ of ~~index <math>p_i^{m_i}</math>~~, ~~and the components are uniquely determined up~~ to ~~isomorphism~~.~~<ref name=GS105>Gille & Szamuely (2006) p.105</ref>~~

	~~==Splitting field==~~
	~~We call a field ''E'' a ''splitting field~~'' for ~~''A'' over ''K'' if ''A''⊗''E'' is isomorphic to a matrix ring over ''E''. Every finite dimensional CSA has a splitting field: indeed~~, in ~~the case when ''A'' is a division algebra, then a maximal subfield of ''A'' is a splitting field. In general there is a splitting field which is a [[separable extension]] of ''K'' of degree equal~~ to ~~the index of ''A''~~, and ~~this splitting field is isomorphic to a subfield of ''A''.<ref name=GS101>Gille & Szamuely~~ (~~2006) p.101</ref> As an example~~, ~~the field '''C''' splits the quaternion algebra '''H''' over '''R''' with~~

	~~:<math> t + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \leftrightarrow~~
	~~\left({\begin{array}{*{20}c} t + x i & y + z i \\ -y + z i & t - x i \end{array}}\right~~) . ~~</math>~~

	~~We can use the existence of the splitting field~~ to ~~define '''reduced norm''' and '''reduced trace'''~~ for ~~a CSA ''A''~~.<~~ref name=GS378~~>~~Gille & Szamuely (2006) pp.37-38~~<~~/ref~~> ~~Map ''A'' to~~ a ~~matrix ring over a splitting field and define the reduced norm and trace~~ to be the ~~composite~~ of ~~this map with determinant and trace respectively~~. ~~For example~~, ~~in the quaternion algebra '''H'''~~, the splitting above shows that the element ''t'' + ''x'' '''i''' + ''y'' '''j''' + ''z'' '''k''' has reduced norm ''t''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> and reduced trace 2''t''.

	~~The reduced norm is multiplicative~~ and the ~~reduced trace is additive~~. ~~An element ''a'' of ''A''~~ is ~~invertible if and only if its reduced norm~~ in ~~non-zero: hence a CSA is a division algebra if~~ and ~~only if the reduced norm is non-zero on the non-zero elements~~.~~<ref name=GS38>Gille & Szamuely (2006) p.38</ref>~~

	~~== Generalization ==~~
	~~CSAs over a field ''K'' are a non-commutative analog~~ to ~~[[extension field]]s over ''K'' – in both cases, they~~ have ~~no non-trivial 2-sided ideals~~, and ~~have a distinguished field in their center~~, ~~though a CSA can be non-commutative and need not have inverses (need not be a~~ [~~[division algebra]])~~. ~~This is of particular interest in [[noncommutative number theory~~]~~] as generalizations of [[number field]]s (extensions of~~ the ~~rationals '''Q'''); see [[noncommutative number field]]~~.

	~~== See also ==~~
	* [[Azumaya algebra]], generalization of CSAs where the ~~base field is replaced by a commutative local ring~~
	* [[Severi–Brauer variety]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| title=Further Algebra and ~~Applications \| first=P.M. \| last=Cohn \| authorlink=Paul Cohn \| edition=2nd \| publisher=Springer \| year=2003 \| isbn=1852336676 \| zbl=1006.00001 }}~~
	* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}
	* {{cite book \| first=Falko \| last=Lorenz \| title=Algebra. Volume II: Fields with ~~Structure, Algebras and Advanced Topics \| year=2008 \| publisher=Springer \| isbn=978-0-387-72487-4 \| zbl=1130~~.~~12001 }}~~

	~~===Further reading===~~
	* {{cite book \| title=Structure of Algebras \| volume=24 \| series=Colloquium Publications \| first=A.A. \| last=Albert \| authorlink=Abraham Adrian Albert \| edition=7th revised reprint \| publisher=American Mathematical Society \| year=1939 \| isbn=0-8218-1024-3 \| zbl=0023.19901 }}
	* {{cite book \| last1=Gille \| first1=Philippe \| last2=Szamuely \| first2=Tamás \| title=Central simple algebras and Galois cohomology \| series=Cambridge Studies in Advanced Mathematics \| volume=101 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2006 \| isbn=0-521-86103-9 \| zbl=1137.~~12001 }}~~

	~~[[Category:Algebras]]~~
	~~[[Category:Ring theory]]~~

en>D.Lazard: Reverted 1 edit by 97.77.5.226 (talk) to last revision by ClueBot NG. (TW)

2014-01-10T16:44:51Z

Reverted 1 edit by 97.77.5.226 (talk) to last revision by ClueBot NG. (TW)

← Older revision		Revision as of 18:44, 10 January 2014
Line 1:		Line 1:
	~~Content material to meet you! A few name is Eusebio Ledbetter. To drive is~~ a ~~complete thing that I~~'~~m completely~~ [~~https://www.google.com/search?hl=en&gl=us&tbm=nws&q=addicted&btnI=lucky addicted~~] ~~to. South Carolina~~ is ~~where options home~~ is and ~~I don~~'~~t plan on transferring it~~. I used to be [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=unemployed&Submit=Go unemployed] but actually I am a ~~cashier but the promotion absolutely not comes. I'm not good at webdesign but you might wish for to check my website: http://circuspartypanama~~.~~com<br><br>~~		In [[ring theory]] and related areas of [[mathematics]] a '''central simple algebra''' ('''CSA''') over a [[field (mathematics)\|field]] ''K'' is a finite-dimensional [[associative algebra]] ''A'', which is [[simple algebra\|simple]], and for which the [[center of an algebra\|center]] is exactly ''K''. In other words, any simple algebra is a central simple algebra over its center.

	~~my webpage~~: ~~clash~~ of ~~clans hack android~~ - [~~http~~://~~circuspartypanama~~.~~com simply click~~ for ~~source~~],		For example, the [[complex number]]s '''C''' form a CSA over themselves, but not over the [[real number]]s '''R''' (the center of '''C''' is all of '''C''', not just '''R'''). The [[quaternion]]s '''H''' form a 4 dimensional CSA over '''R''', and in fact represent the only non-trivial element of the [[Brauer group]] of the reals (see below).

			Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''[[Brauer equivalent]]'') if their division rings ''S'' and ''T'' are isomorphic. The set of all [[equivalence class]]es of central simple algebras over a given field ''F'', under this equivalence relation, can be equipped with a [[group operation]] given by the [[tensor product of algebras]]. The resulting group is called the [[Brauer group]] Br(''F'') of the field ''F''.<ref name=L159>Lorenz (2008) p.159</ref> It is always a [[torsion group]].<ref name=L194>Lorenz (2008) p.194</ref>

			== Properties ==
			* According to the [[Artin–Wedderburn theorem]] a finite-dimensional simple algebra ''A'' is isomorphic to the matrix algebra [[matrix ring\|''M''(''n'',''S'')]] for some [[division ring]] ''S''. Hence, there is a unique division algebra in each Brauer equivalence class.<ref name=L160>Lorenz (2008) p.160</ref>
			* Every [[automorphism]] of a central simple algebra is an [[inner automorphism]] (follows from [[Skolem–Noether theorem]]).
			* The dimension of a central simple algebra as a vector space over its centre is always a square: the ''degree'' is the square root of this dimension. The '''Schur index''' of a central simple algebra is the degree of the equivalent division algebra:<ref name=L163>Lorenz (2008) p.163</ref> it depends only on the [[Brauer class]] of the algebra.<ref name=GS100>Gille & Szamuely (2006) p.100</ref>
			* The ''period'' of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index, and the two numbers are composed of the same prime factors.<ref name=GS104>Gille & Szamuely (2006) p.104</ref>
			* If ''S'' is a simple [[subalgebra]] of a central simple algebra ''A'' then dim<sub>''F''</sub> ''S'' divides dim<sub>''F''</sub> ''A''.
			* Every 4-dimensional central simple algebra over a field ''F'' is isomorphic to a [[quaternion algebra]]; in fact, it is either a two-by-two [[matrix algebra]], or a [[division algebra]].
			* If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation
			::<math>\mathrm{ind}(D) = \prod_{i=1}^r p_i^{m_i} \ </math>
			:then ''D'' has a tensor product decomposition
			::<math>D = \otimes_{i=1}^r D_i \ </math>
			:where each component ''D''<sub>''i''</sub> is a central division algebra of index <math>p_i^{m_i}</math>, and the components are uniquely determined up to isomorphism.<ref name=GS105>Gille & Szamuely (2006) p.105</ref>

			==Splitting field==
			We call a field ''E'' a ''splitting field'' for ''A'' over ''K'' if ''A''⊗''E'' is isomorphic to a matrix ring over ''E''. Every finite dimensional CSA has a splitting field: indeed, in the case when ''A'' is a division algebra, then a maximal subfield of ''A'' is a splitting field. In general there is a splitting field which is a [[separable extension]] of ''K'' of degree equal to the index of ''A'', and this splitting field is isomorphic to a subfield of ''A''.<ref name=GS101>Gille & Szamuely (2006) p.101</ref> As an example, the field '''C''' splits the quaternion algebra '''H''' over '''R''' with

			:<math> t + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \leftrightarrow
			\left({\begin{array}{*{20}c} t + x i & y + z i \\ -y + z i & t - x i \end{array}}\right) . </math>

			We can use the existence of the splitting field to define '''reduced norm''' and '''reduced trace''' for a CSA ''A''.<ref name=GS378>Gille & Szamuely (2006) pp.37-38</ref> Map ''A'' to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebra '''H''', the splitting above shows that the element ''t'' + ''x'' '''i''' + ''y'' '''j''' + ''z'' '''k''' has reduced norm ''t''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> and reduced trace 2''t''.

			The reduced norm is multiplicative and the reduced trace is additive. An element ''a'' of ''A'' is invertible if and only if its reduced norm in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.<ref name=GS38>Gille & Szamuely (2006) p.38</ref>

			== Generalization ==
			CSAs over a field ''K'' are a non-commutative analog to [[extension field]]s over ''K'' – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a [[division algebra]]). This is of particular interest in [[noncommutative number theory]] as generalizations of [[number field]]s (extensions of the rationals '''Q'''); see [[noncommutative number field]].

			== See also ==
			* [[Azumaya algebra]], generalization of CSAs where the base field is replaced by a commutative local ring
			* [[Severi–Brauer variety]]

			==References==
			{{reflist}}
			* {{cite book \| title=Further Algebra and Applications \| first=P.M. \| last=Cohn \| authorlink=Paul Cohn \| edition=2nd \| publisher=Springer \| year=2003 \| isbn=1852336676 \| zbl=1006.00001 }}
			* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}
			* {{cite book \| first=Falko \| last=Lorenz \| title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics \| year=2008 \| publisher=Springer \| isbn=978-0-387-72487-4 \| zbl=1130.12001 }}

			===Further reading===
			* {{cite book \| title=Structure of Algebras \| volume=24 \| series=Colloquium Publications \| first=A.A. \| last=Albert \| authorlink=Abraham Adrian Albert \| edition=7th revised reprint \| publisher=American Mathematical Society \| year=1939 \| isbn=0-8218-1024-3 \| zbl=0023.19901 }}
			* {{cite book \| last1=Gille \| first1=Philippe \| last2=Szamuely \| first2=Tamás \| title=Central simple algebras and Galois cohomology \| series=Cambridge Studies in Advanced Mathematics \| volume=101 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2006 \| isbn=0-521-86103-9 \| zbl=1137.12001 }}

			[[Category:Algebras]]
			[[Category:Ring theory]]

en>YFdyh-bot: r2.7.3) (Robot: Adding cs:Polynomická rovnice

2012-08-18T20:42:06Z

r2.7.3) (Robot: Adding cs:Polynomická rovnice

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