Intersection (set theory) - Revision history

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2014-11-19T09:30:05Z

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The Knightsbridge, Swell Tall, Classic Short, and Elijo all feature~~ a ~~double-layer~~ of ~~sheepskin in~~ the ~~lining to provide exceptional warmth and to help keep your feet dry~~. ~~The soles~~ of ~~these boots are designed~~ to ~~provide superior traction, even in slick ice and snow~~. As with all [http://tinyurl.com/ku6vjks ugg boots sale] boots, these winter weather boots are not only functional, but they are designed with style and fashion in mind. So there's no need to sacrifice warmth and comfort for ~~style. <br><br>No matter what kind of women's boots you're looking for, UGG has a vast selection~~ of ~~comfortable, stylish options that~~ will ~~keep~~ you ~~and your feet happy all year~~.		You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. Infertility can cause a major setback to the couples due to the inability to conceive. This plugin is a must have for anyone who is serious about using Word - Press. 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94.189.147.2 at 18:13, 15 February 2014

2014-02-15T18:13:51Z

← Older revision		Revision as of 20:13, 15 February 2014
Line 1:		Line 1:
	~~In [[algebraic number theory]], the~~ '~~''Albert–Brauer–Hasse–Noether theorem''' states that~~ a [~~[central simple algebra]] over an [[algebraic number field]] ''K'' which splits over every [[local field\|completion]] ''K''<sub>''v''<~~/~~sub> is a [[matrix ring\|matrix algebra~~]~~] over ''K'~~'. ~~The theorem is an example of a [[local-global principle]] in [[algebraic number theory]] and~~		Shopping for women's boots can be frustrating. Some brands only offer a few styles of boots for women, [http://Data.gov.uk/data/search?q=leading leading] most women to think that their boot options are limited. And if you're in the market for warm weather boots, your options can seem even slimmer. But when shopping for UGG boots, many women are surprised to find that there are a number of different styles to choose from, all of which are designed to be fashionable and warm. <br><br>
	~~leads~~ to a ~~complete description of finite-dimensional [[division algebras]] over algebraic~~ number ~~fields in terms~~ of ~~their [[Hasse invariant~~ of ~~an algebra\|local invariants]]. It was proved independently by [[Helmut Hasse]], [[Richard Brauer]],~~ and ~~[[Emmy Noether]] and by [[Abraham Adrian Albert]] and Hasse~~.

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	~~Let~~ '~~'A'' be a~~ [~~[central simple algebra]~~] of ~~rank ''d'' over an~~ [~~[algebraic number field~~]~~] ''K''~~. ~~Suppose that for any [[valuation (algebra)\|valuation]] ''v'', ''A'' splits over the corresponding local field ''K''~~<~~sub~~>~~''v''~~<~~/sub~~>:

	: <~~math~~> ~~A\otimes_K K_v \simeq M_d(K_v).~~ <~~/math~~>

	~~Then ''A''~~ is ~~isomorphic~~ to the ~~matrix algebra ''M~~''<~~sub~~>~~''d''~~<~~/sub~~>~~(''K'').~~

	~~== Applications ==~~

	~~Using~~ the ~~theory of [[Brauer group]]~~, ~~one shows that two central simple algebras ''A''~~ and ~~''B'' over~~ an ~~algebraic number field ''K'' are isomorphic over ''K'' if and only if their completions ''A''~~<~~sub~~>~~''v''~~<~~/sub~~> ~~and ''B''~~<~~sub~~>~~''v''~~<~~/sub~~> are ~~isomorphic over the completion ''K''<sub>''v''</sub> for every ''v''.~~

	~~Together with the [[Grunwald–Wang theorem]], the Albert–Brauer–Hasse–Noether theorem implies~~ that ~~every central simple algebra over an algebraic number field is ''cyclic''~~, i.e. ~~can be obtained by an explicit construction from a [[cyclic extension\|cyclic field extension]] ''L''~~/~~''K''~~ .

	== ~~See also ==~~

	* [[Class field theory]]

	* [[Hasse norm theorem]]

	~~==References==~~

	* {{citation \| author1-link=Abraham Adrian Albert \| first1=A.A. \| last1=Albert \| author2-link=Helmut Hasse \| first2=H. ~~\| last2=Hasse \| title=A determination~~ of all ~~normal division algebras over an algebraic number field \| journal=Trans~~. ~~Amer~~. ~~Math~~. ~~Soc~~. ~~\| volume=34 \| year=1932 \| pages=722–726 \| zbl=0005~~.~~05003 }}~~
	* {{citation \| author1-link=Helmut Hasse \| first1=H. \| last1=Hasse \| author2-link=Richard Brauer \| first2=R. \| last2=Brauer \| author3-link=Emmy Noether \| first3=E. \| last3=Noether \| title=Beweis eines Hauptsatzes in ~~der Theorie der Algebren \| journal=Journal für Mathematik \| volume=167 \| year=1931 \| pages=399–404 }}~~
	*{{Citation
	~~\| doi = 10~~.~~1007/s00407-004-0093-6~~
	~~\| title = Delicate collaboration: Adrian Albert~~ and ~~Helmut Hasse~~ and the ~~Principal Theorem in Division Algebras~~
	~~\| url =~~ http://~~www.springerlink~~.com/~~index/YLLATB4EMBEG6Q3H~~.~~pdf~~
	~~\| year = 2005~~
	~~\| journal = Archive for history~~ of ~~exact sciences~~
	~~\| pages = 349–379~~
	~~\| volume = 59~~
	~~\| issue = 4~~
	~~\| last1 = Fenster \| first1 = D~~.D.
	~~\| last2 = Schwermer \| first2 = J~~.
	~~\| accessdate = 2009-07-05 }}~~
	* {{citation \| first=Richard \| last=Pierce \| title=Associative algebras \| series=Graduate Texts in ~~Mathematics \| volume=88 \| series=Studies~~ in the ~~History~~ of ~~Modern Science \| volume=9 \| publisher=[[Springer-Verlag]] \| location=New York-Berlin \| year=1982 \| isbn=0~~-~~387-90693-2 \| zbl=0497~~.~~16001 }}~~
	* {{citation \| last=Reiner \| first=I. ~~\| authorlink=Irving Reiner \| title=Maximal Orders \| series=London Mathematical Society Monographs. New Series \| volume=28 \| publisher=[[Oxford University Press~~]~~] \|~~ year~~=2003 \| isbn=0-19-852673-3 \| zbl=1024~~.~~16008 \| page=276 }}~~
	*{{Citation
	~~\| title =~~ The ~~Brauer–Hasse–Noether theorem~~ in ~~historical perspective~~
	~~\| id = {{citeseerx\|10~~.~~1.1.72.4101}}~~
	~~\| year = 2005~~
	~~\|journal=Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften~~
	~~\| last= Roquette~~
	~~\|first= Peter~~
	~~\|volume=15~~
	~~\| mr=2222818 \| zbl=1060.01009~~
	~~\| accessdate = 2009-07-05~~
	~~Albert~~, ~~Nancy E~~. ~~(2005), "A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8~~
	}}

	~~== Notes ==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Albert-Brauer-Hasse-Noether theorem}}~~
	[~~[Category~~:~~Class field theory~~]]
	~~[[Category:Theorems in algebraic number theory]]~~
	~~Please add~~, in ~~the References~~, a ~~reference to Chapter 10 in the biography "A Cubed & His Algebra~~,~~" which is devoted to discussion of the Albert-Brauer-Hasse-Noether theorem~~.

en>Paul August: Reverted edits by 150.176.167.224 (talk) to last version by Mate2code

2013-11-14T01:24:43Z

Reverted edits by 150.176.167.224 (talk) to last version by Mate2code

← Older revision		Revision as of 03:24, 14 November 2013
Line 1:		Line 1:
			In [[algebraic number theory]], the '''Albert–Brauer–Hasse–Noether theorem''' states that a [[central simple algebra]] over an [[algebraic number field]] ''K'' which splits over every [[local field\|completion]] ''K''<sub>''v''</sub> is a [[matrix ring\|matrix algebra]] over ''K''. The theorem is an example of a [[local-global principle]] in [[algebraic number theory]] and
			leads to a complete description of finite-dimensional [[division algebras]] over algebraic number fields in terms of their [[Hasse invariant of an algebra\|local invariants]]. It was proved independently by [[Helmut Hasse]], [[Richard Brauer]], and [[Emmy Noether]] and by [[Abraham Adrian Albert]] and Hasse.

			== Statement of the theorem ==

	~~CMS provides the best platform to create websites~~ that ~~fulfill all~~ the ~~specifications of SEO~~. ~~The next step~~ is to ~~visit your Word - Press blog dashboard~~. ~~Wordpress Content management systems~~, ~~being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not~~ that ~~easy because of the simplified way in creating your very own themes. All this is very~~ simple, and ~~the best thing is that it is totally free,~~ and ~~you don~~'~~t need a domain name or web hosting.~~ <br><br>~~purcase~~ and ~~download - WPZOOM Tribune wordpress Theme, find and use~~ the ~~WPZOOM Discount Code~~. ~~Best of all~~, ~~you can still have all~~ the ~~functionality~~ that ~~you desire when you use the Word~~ - ~~Press platform~~. ~~Well Managed Administration The Word~~ - ~~Press can easily absorb the high numbers~~ of ~~traffic by controlling the server load to make sure that the site works properly~~. ~~Apart from these, you are also required to give some backlinks on other sites as well~~. ~~But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help~~. ~~<br><br>This gives a clearer picture that online shoppers are familiar with the WP ecommerce system~~. ~~By using Word~~ - ~~Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply~~. ~~We can active Akismet from wp~~-~~admin > Plugins > Installed Plugins~~. ~~Thousands of plugins are available~~ in ~~Word~~ - ~~Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. If you have any questions on starting a Word~~ - ~~Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>It has become a more prevalent cause of infertility~~ and ~~the fertility clinic are having more~~ and ~~more couples with infertility problems. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst~~ the ~~best~~ in ~~the world,and have been for some time.~~ ~~Here's more information on [~~http://~~urlon~~.com.~~br/wordpress_backup_843900 wordpress dropbox backup~~] ~~look into the internet site~~. ~~Websites that do rank highly, do so becaue they use keyword~~-~~heavy post titles~~. ~~If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice~~. ~~If your site does well you can get paid professional designer to create a unique Word~~ - ~~Press theme~~. ~~<br><br>He loves sharing information regarding wordpress~~, ~~Majento~~, ~~Drupal and Joomla development tips~~ & ~~tricks. As a website owner~~, ~~you can easily manage CMS~~-~~based website~~ in a ~~pretty easy and convenient style. This allows updates~~ to ~~be sent anyone who wants them via an RSS reader or directly to their email. This is because of~~ the ~~customization that works as a keystone for a SEO friendly blogging portal website. As with a terminology~~, ~~there are many methods~~ to ~~understand how to use~~ the ~~terminology~~.		Let ''A'' be a [[central simple algebra]] of rank ''d'' over an [[algebraic number field]] ''K''. Suppose that for any [[valuation (algebra)\|valuation]] ''v'', ''A'' splits over the corresponding local field ''K''<sub>''v''</sub>:

			: <math> A\otimes_K K_v \simeq M_d(K_v). </math>

			Then ''A'' is isomorphic to the matrix algebra ''M''<sub>''d''</sub>(''K'').

			== Applications ==

			Using the theory of [[Brauer group]], one shows that two central simple algebras ''A'' and ''B'' over an algebraic number field ''K'' are isomorphic over ''K'' if and only if their completions ''A''<sub>''v''</sub> and ''B''<sub>''v''</sub> are isomorphic over the completion ''K''<sub>''v''</sub> for every ''v''.

			Together with the [[Grunwald–Wang theorem]], the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is ''cyclic'', i.e. can be obtained by an explicit construction from a [[cyclic extension\|cyclic field extension]] ''L''/''K'' .

			== See also ==

			* [[Class field theory]]

			* [[Hasse norm theorem]]

			==References==

			* {{citation \| author1-link=Abraham Adrian Albert \| first1=A.A. \| last1=Albert \| author2-link=Helmut Hasse \| first2=H. \| last2=Hasse \| title=A determination of all normal division algebras over an algebraic number field \| journal=Trans. Amer. Math. Soc. \| volume=34 \| year=1932 \| pages=722–726 \| zbl=0005.05003 }}
			* {{citation \| author1-link=Helmut Hasse \| first1=H. \| last1=Hasse \| author2-link=Richard Brauer \| first2=R. \| last2=Brauer \| author3-link=Emmy Noether \| first3=E. \| last3=Noether \| title=Beweis eines Hauptsatzes in der Theorie der Algebren \| journal=Journal für Mathematik \| volume=167 \| year=1931 \| pages=399–404 }}
			*{{Citation
			\| doi = 10.1007/s00407-004-0093-6
			\| title = Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras
			\| url = http://www.springerlink.com/index/YLLATB4EMBEG6Q3H.pdf
			\| year = 2005
			\| journal = Archive for history of exact sciences
			\| pages = 349–379
			\| volume = 59
			\| issue = 4
			\| last1 = Fenster \| first1 = D.D.
			\| last2 = Schwermer \| first2 = J.
			\| accessdate = 2009-07-05 }}
			* {{citation \| first=Richard \| last=Pierce \| title=Associative algebras \| series=Graduate Texts in Mathematics \| volume=88 \| series=Studies in the History of Modern Science \| volume=9 \| publisher=[[Springer-Verlag]] \| location=New York-Berlin \| year=1982 \| isbn=0-387-90693-2 \| zbl=0497.16001 }}
			* {{citation \| last=Reiner \| first=I. \| authorlink=Irving Reiner \| title=Maximal Orders \| series=London Mathematical Society Monographs. New Series \| volume=28 \| publisher=[[Oxford University Press]] \| year=2003 \| isbn=0-19-852673-3 \| zbl=1024.16008 \| page=276 }}
			*{{Citation
			\| title = The Brauer–Hasse–Noether theorem in historical perspective
			\| id = {{citeseerx\|10.1.1.72.4101}}
			\| year = 2005
			\|journal=Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften
			\| last= Roquette
			\|first= Peter
			\|volume=15
			\| mr=2222818 \| zbl=1060.01009
			\| accessdate = 2009-07-05
			Albert, Nancy E. (2005), "A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8
			}}

			== Notes ==
			{{reflist}}

			{{DEFAULTSORT:Albert-Brauer-Hasse-Noether theorem}}
			[[Category:Class field theory]]
			[[Category:Theorems in algebraic number theory]]
			Please add, in the References, a reference to Chapter 10 in the biography "A Cubed & His Algebra," which is devoted to discussion of the Albert-Brauer-Hasse-Noether theorem.

27.58.81.66: /* Intersecting and disjoint sets */

2012-07-21T08:50:48Z

Intersecting and disjoint sets

New page

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← Older revision		Revision as of 20:13, 15 February 2014
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	~~In [[algebraic number theory]], the~~ '~~''Albert–Brauer–Hasse–Noether theorem''' states that~~ a [~~[central simple algebra]] over an [[algebraic number field]] ''K'' which splits over every [[local field\|completion]] ''K''<sub>''v''<~~/~~sub> is a [[matrix ring\|matrix algebra~~]~~] over ''K'~~'. ~~The theorem is an example of a [[local-global principle]] in [[algebraic number theory]] and~~		Shopping for women's boots can be frustrating. Some brands only offer a few styles of boots for women, [http://Data.gov.uk/data/search?q=leading leading] most women to think that their boot options are limited. And if you're in the market for warm weather boots, your options can seem even slimmer. But when shopping for UGG boots, many women are surprised to find that there are a number of different styles to choose from, all of which are designed to be fashionable and warm. <br><br>
	~~leads~~ to a ~~complete description of finite-dimensional [[division algebras]] over algebraic~~ number ~~fields in terms~~ of ~~their [[Hasse invariant~~ of ~~an algebra\|local invariants]]. It was proved independently by [[Helmut Hasse]], [[Richard Brauer]],~~ and ~~[[Emmy Noether]] and by [[Abraham Adrian Albert]] and Hasse~~.

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	~~Let~~ '~~'A'' be a~~ [~~[central simple algebra]~~] of ~~rank ''d'' over an~~ [~~[algebraic number field~~]~~] ''K''~~. ~~Suppose that for any [[valuation (algebra)\|valuation]] ''v'', ''A'' splits over the corresponding local field ''K''~~<~~sub~~>~~''v''~~<~~/sub~~>:

	: <~~math~~> ~~A\otimes_K K_v \simeq M_d(K_v).~~ <~~/math~~>

	~~Then ''A''~~ is ~~isomorphic~~ to the ~~matrix algebra ''M~~''<~~sub~~>~~''d''~~<~~/sub~~>~~(''K'').~~

	~~== Applications ==~~

	~~Using~~ the ~~theory of [[Brauer group]]~~, ~~one shows that two central simple algebras ''A''~~ and ~~''B'' over~~ an ~~algebraic number field ''K'' are isomorphic over ''K'' if and only if their completions ''A''~~<~~sub~~>~~''v''~~<~~/sub~~> ~~and ''B''~~<~~sub~~>~~''v''~~<~~/sub~~> are ~~isomorphic over the completion ''K''<sub>''v''</sub> for every ''v''.~~

	~~Together with the [[Grunwald–Wang theorem]], the Albert–Brauer–Hasse–Noether theorem implies~~ that ~~every central simple algebra over an algebraic number field is ''cyclic''~~, i.e. ~~can be obtained by an explicit construction from a [[cyclic extension\|cyclic field extension]] ''L''~~/~~''K''~~ .

	== ~~See also ==~~

	* [[Class field theory]]

	* [[Hasse norm theorem]]

	~~==References==~~

	* {{citation \| author1-link=Abraham Adrian Albert \| first1=A.A. \| last1=Albert \| author2-link=Helmut Hasse \| first2=H. ~~\| last2=Hasse \| title=A determination~~ of all ~~normal division algebras over an algebraic number field \| journal=Trans~~. ~~Amer~~. ~~Math~~. ~~Soc~~. ~~\| volume=34 \| year=1932 \| pages=722–726 \| zbl=0005~~.~~05003 }}~~
	* {{citation \| author1-link=Helmut Hasse \| first1=H. \| last1=Hasse \| author2-link=Richard Brauer \| first2=R. \| last2=Brauer \| author3-link=Emmy Noether \| first3=E. \| last3=Noether \| title=Beweis eines Hauptsatzes in ~~der Theorie der Algebren \| journal=Journal für Mathematik \| volume=167 \| year=1931 \| pages=399–404 }}~~
	*{{Citation
	~~\| doi = 10~~.~~1007/s00407-004-0093-6~~
	~~\| title = Delicate collaboration: Adrian Albert~~ and ~~Helmut Hasse~~ and the ~~Principal Theorem in Division Algebras~~
	~~\| url =~~ http://~~www.springerlink~~.com/~~index/YLLATB4EMBEG6Q3H~~.~~pdf~~
	~~\| year = 2005~~
	~~\| journal = Archive for history~~ of ~~exact sciences~~
	~~\| pages = 349–379~~
	~~\| volume = 59~~
	~~\| issue = 4~~
	~~\| last1 = Fenster \| first1 = D~~.D.
	~~\| last2 = Schwermer \| first2 = J~~.
	~~\| accessdate = 2009-07-05 }}~~
	* {{citation \| first=Richard \| last=Pierce \| title=Associative algebras \| series=Graduate Texts in ~~Mathematics \| volume=88 \| series=Studies~~ in the ~~History~~ of ~~Modern Science \| volume=9 \| publisher=[[Springer-Verlag]] \| location=New York-Berlin \| year=1982 \| isbn=0~~-~~387-90693-2 \| zbl=0497~~.~~16001 }}~~
	* {{citation \| last=Reiner \| first=I. ~~\| authorlink=Irving Reiner \| title=Maximal Orders \| series=London Mathematical Society Monographs. New Series \| volume=28 \| publisher=[[Oxford University Press~~]~~] \|~~ year~~=2003 \| isbn=0-19-852673-3 \| zbl=1024~~.~~16008 \| page=276 }}~~
	*{{Citation
	~~\| title =~~ The ~~Brauer–Hasse–Noether theorem~~ in ~~historical perspective~~
	~~\| id = {{citeseerx\|10~~.~~1.1.72.4101}}~~
	~~\| year = 2005~~
	~~\|journal=Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften~~
	~~\| last= Roquette~~
	~~\|first= Peter~~
	~~\|volume=15~~
	~~\| mr=2222818 \| zbl=1060.01009~~
	~~\| accessdate = 2009-07-05~~
	~~Albert~~, ~~Nancy E~~. ~~(2005), "A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8~~
	}}

	~~== Notes ==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Albert-Brauer-Hasse-Noether theorem}}~~
	[~~[Category~~:~~Class field theory~~]]
	~~[[Category:Theorems in algebraic number theory]]~~
	~~Please add~~, in ~~the References~~, a ~~reference to Chapter 10 in the biography "A Cubed & His Algebra~~,~~" which is devoted to discussion of the Albert-Brauer-Hasse-Noether theorem~~.