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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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| 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''.
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| 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.
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| == Witt's cancellation theorem ==
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| 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
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| : <math> (V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q).</math>
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| Then the quadratic spaces (''V''<sub>1</sub>, ''q''<sub>1</sub>) and (''V''<sub>2</sub>, ''q''<sub>2</sub>) are isometric:
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| : <math> (V_1,q_1)\simeq (V_2,q_2).</math>
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| In other words, the direct summand (''V'', ''q'') appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".
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| == Witt's decomposition theorem ==
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| Let (''V'', ''q'') be a quadratic space over a field ''k''. Then
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| it admits a '''Witt decomposition''':
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| : <math>(V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h),</math>
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| 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>
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| Quadratic forms with the same core form are said to be ''similar'' or '''Witt equivalent'''.
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| == References == | |
| {{reflist}}
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| * {{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 }}
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| * {{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 }}
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| [[Category:Theorems in algebra]]
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| [[Category:Quadratic forms]]
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