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2014-11-03T17:15:00Z

Journal cites, added 1 PMID using AWB (10484)

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en>Wikid77: 28 changes: ce; fix 2 cite "comment=7" & 8 "coauthors=" as "author2="

2014-02-20T20:46:51Z

28 changes: ce; fix 2 cite "comment=7" & 8 "coauthors=" as "author2="

← Older revision		Revision as of 22:46, 20 February 2014
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	~~:''"Witt's theorem" or "the Witt theorem" may also refer to the [[Bourbaki–Witt theorem\|Bourbaki–Witt fixed point theorem]] of order theory.''~~		Andera is what you can call her but she never truly liked that title. What me and my family love is bungee leaping but I've been taking on new issues recently. My wife and I reside in Kentucky. Credit authorising is how he tends to make cash.<br><br>my website ... tarot readings - [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ click the following page],

	~~In mathematics, '''Witt's theorem''', named after [[Ernst Witt]],~~ is a basic result in the algebraic theory of [[quadratic form]]s: any [[Isometry (quadratic forms)\|isometry]] between two subspaces of a nonsingular [[quadratic space]] over a [[field (algebra)\|field]] ''k'' may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian [[bilinear form]]s over arbitrary fields. ~~The theorem applies to classification of quadratic forms over ''k''~~ and ~~in particular allows one to define the [[Witt group]] ''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''.~~

	~~== Statement of the theorem ==~~

	Let (''V'', ''b'') be a finite-dimensional vector space over an arbitrary [[field (algebra)\|field]] ''k'' together with a nondegenerate symmetric or skew-symmetric [[bilinear form]]. If ''f'': ''U''→''U' '' is an [[isometry]] between two subspaces of ''V'' then ''f'' extends to an isometry of ''V''.

	~~Witt's theorem implies that the dimension of a maximal [[isotropic subspace]] of ''V''~~ is ~~an invariant, called the '''index''~~' ~~or '''{{visible anchor\|Witt index}}''' of ''b'', and moreover, that the [[isometry group]] of (''V'', ''b'') [[group action\|acts]] transitively~~ on ~~the set of maximal isotropic subspaces~~. ~~This fact plays an important role in the structure theory and [[group representation\|representation theory]] of the isometry group~~ and in ~~the theory of [[reductive dual pair]]s~~.

	~~== Witt's cancellation theorem ==~~

	~~Let (''V'', ''q''), (''V''<sub>1</sub>, ''q''<sub>1</sub>), (''V''<sub>2</sub>, ''q''<sub>2</sub>) be three quadratic spaces over a field ''k''~~. ~~Assume that~~

	~~: <math> (V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q).</math>~~

	~~Then the quadratic spaces (''V''~~<~~sub~~>1<~~/sub~~>~~, ''q''<sub>1</sub>) and (''V''<sub>2</sub>, ''q''<sub>2</sub>) are isometric:~~

	~~: <math> (V_1,q_1)\simeq (V_2,q_2)~~.~~</math>~~

	~~In other words, the direct summand (''V'', ''q'') appearing in both sides of an isomorphism between quadratic spaces may be "cancelled"~~.

	~~== Witt's decomposition theorem ==~~

	~~Let (''V'', ''q'') be a quadratic space over a field ''k''~~. ~~Then~~
	~~it admits a '''Witt decomposition'''~~:

	~~: <math>(V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h),<~~/~~math>~~

	~~where ''V''<sub>0<~~/~~sub>=ker ''q'' is the [[Radical of a quadratic space\|radical]] of ''q'', (''V''<sub>''a''</sub>, ''q''<sub>''a''<~~/sub>) is an [[anisotropic quadratic space]] and (''V''<sub>''h''</sub>, ''q''<sub>''h''</sub>) is a [[split quadratic space]]. Moreover, the anisotropic summand, termed the '''core form''', and the hyperbolic summand in a Witt decomposition of (''V'', ''q'') are determined uniquely up to isomorphism.<ref>Lorenz (2008) p.30</~~ref>~~

	~~Quadratic forms with the same core form are said to be ''similar'' or '''Witt equivalent'''.~~

	~~== References ==~~
	~~{{reflist}}~~
	* {{cite book \| first=O. Timothy \| last=O'Meara \| authorlink=O. Timothy O'Meara \| title=Introduction to Quadratic Forms \| publisher=[[Springer-~~Verlag]] \| year=1973 \| series=Die Grundlehren der mathematischen Wissenschaften \| volume=117 \| zbl=0259.10018 }}~~
	* {{cite book \| first=Falko \| last=Lorenz \| title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics \| year=2008 \| publisher=[[Springer-~~Verlag]] \| isbn=978~~-0-~~387~~-~~72487~~-~~4 \| pages=15–27 \| zbl=1130.12001 }}~~

	~~[[Category:Theorems in algebra]]~~
	~~[[Category:Quadratic forms]~~]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4117379

2013-03-15T07:21:32Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q4117379

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+:''"Witt's theorem" or "the Witt theorem" may also refer to the [[Bourbaki–Witt theorem|Bourbaki–Witt fixed point theorem]] of order theory.''
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+In mathematics, '''Witt's theorem''', named after [[Ernst Witt]], is a basic result in the algebraic theory of [[quadratic form]]s:  any [[Isometry (quadratic forms)|isometry]] between two subspaces of a nonsingular [[quadratic space]] over a [[field (algebra)|field]] ''k'' may be extended to an isometry of the whole space.  An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian [[bilinear form]]s over arbitrary fields. The theorem applies to classification of quadratic forms over ''k'' and in particular allows one to define the [[Witt group]] ''W''(''k'') which describes the "stable" theory of quadratic forms over the field ''k''.
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+== Statement of the theorem ==

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