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of maximal isotropic subspaces. This fact plays an important role in the structure theory and
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may be extended to an isometry of the whole space. An analogous statement holds also for
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over arbitrary fields. The theorem applies to classification of quadratic forms over
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27:
Basic result in the algebraic theory of quadratic forms, on extending isometries
895:
717:{\displaystyle (V,q)\simeq (V_{0},0)\oplus (V_{a},q_{a})\oplus (V_{h},q_{h}),}
96:
46:
420:{\displaystyle (V_{1},q_{1})\oplus (V,q)\simeq (V_{2},q_{2})\oplus (V,q).}
171:
106:) which describes the "stable" theory of quadratic forms over the field
955:
Algebra. Volume II: Fields with
Structure, Algebras and Advanced Topics
1015:, Die Grundlehren der mathematischen Wissenschaften, vol. 117,
32:"Witt's theorem" or "the Witt theorem" may also refer to the
804:, and the hyperbolic summand in a Witt decomposition of
819:
Quadratic forms with the same core form are said to be
189:
Witt's theorem implies that the dimension of a maximal
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543:{\displaystyle (V_{1},q_{1})\simeq (V_{2},q_{2}).}
542:
419:
800:. Moreover, the anisotropic summand, termed the
923:, vol. 67, American Mathematical Society,
569:between quadratic spaces may be "cancelled".
8:
917:Introduction to Quadratic Forms over Fields
816:are determined uniquely up to isomorphism.
243:of the isometry group and in the theory of
95:and in particular allows one to define the
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303:be three quadratic spaces over a field
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553:In other words, the direct summand
589:be a quadratic space over a field
25:
148:different from 2 together with a
34:Bourbaki–Witt fixed point theorem
18:Witt's decomposition theorem
1013:Introduction to Quadratic Forms
921:Graduate Studies in Mathematics
565:appearing in both sides of an
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992:Graduate Texts in Mathematics
994:(Third ed.), Springer,
573:Witt's decomposition theorem
197:is an invariant, called the
152:symmetric or skew-symmetric
772:anisotropic quadratic space
251:Witt's cancellation theorem
49:, is a basic result in the
1062:
1041:Theorems in linear algebra
846:, p. 275-276, ch. 11.
430:Then the quadratic spaces
191:totally isotropic subspace
182:extends to an isometry of
213:, and moreover, that the
174:between two subspaces of
988:Advanced Linear Algebra
953:Lorenz, Falko (2008),
870:, p. 296, ch. 11.
718:
544:
421:
798:split quadratic space
719:
545:
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241:representation theory
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245:reductive dual pairs
1009:O'Meara, O. Timothy
593:. Then it admits a
961:, pp. 15–27,
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595:Witt decomposition
540:
417:
132:finite-dimensional
1001:978-0-387-72828-5
968:978-0-387-72487-4
901:Geometric Algebra
65:of a nonsingular
16:(Redirected from
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193:(null space) of
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41:In mathematics,
36:of order theory.
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67:quadratic space
55:quadratic forms
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307:. Assume that
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215:isometry group
150:non-degenerate
146:characteristic
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89:bilinear forms
86:skew-Hermitian
78:skew-symmetric
45:, named after
43:Witt's theorem
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233:transitively
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135:vector space
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99:
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61:between two
42:
40:
31:
880:Lorenz 2008
567:isomorphism
1035:Categories
1025:0259.10018
977:1130.12001
947:1068.11023
903:, page 121
896:Emil Artin
890:References
868:Roman 2008
844:Roman 2008
205:Witt index
97:Witt group
47:Ernst Witt
831:Citations
802:core form
680:⊕
648:⊕
623:≃
506:≃
397:⊕
365:≃
347:⊕
114:Statement
82:Hermitian
63:subspaces
1011:(1973),
986:(2008),
915:(2005),
856:Lam 2005
172:isometry
161: :
59:isometry
939:2104929
898:(1957)
821:similar
742:radical
740:is the
235:on the
137:over a
69:over a
57:: any
1023:
998:
975:
965:
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937:
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770:is an
735:= ker
727:where
170:is an
796:is a
199:index
178:then
156:. If
139:field
130:be a
71:field
996:ISBN
963:ISBN
925:ISBN
905:via
774:and
577:Let
448:and
255:Let
230:acts
118:Let
84:and
1021:Zbl
973:Zbl
943:Zbl
823:or
744:of
237:set
217:of
209:of
201:or
144:of
53:of
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748:,
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