565:; equivalently, the index of isotropy is equal to half the dimension. The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.
573:
From the point of view of classification of quadratic forms, spaces with definite quadratic forms are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field
751:
577:, classification of definite quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By
905:
871:
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578:
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893:
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46:
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is a quadratic space of dimension at least three, then it is isotropic (this is a consequence of the
603:
812:
582:
205:
35:
397:
has been used by Milnor and
Husemoller for the hyperbolic plane as the signs of the terms of the
901:
881:
867:
844:
808:
759:
709:
911:
797:
769:
201:
87:. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or
915:
897:
863:
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162:
of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.
115:
107:
17:
830:
694:
390:
314:
31:
27:
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
926:
886:
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666:
45:
if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a
630:
54:
743:
699:
322:
88:
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704:
515:
408:
398:
366:
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is a quadratic space of dimension at least five, then it is isotropic.
622:
is a quadratic space of dimension at least two, then it is isotropic.
188:. An important example of an isotropic form over the reals occurs in
165:
More generally, if the quadratic form is non-degenerate and has the
896:: Classics in mathematics. Vol. 7 (reprint of 3rd ed.).
589:
of a split space and an space with definite quadratic form.
438:, where the products represent the quadratic form.
561:) if there is a subspace which is equal to its own
885:
752:Ergebnisse der Mathematik und ihrer Grenzgebiete
829:, §1.3 Hyperbolic plane and hyperbolic spaces,
569:Relation with classification of quadratic forms
407:The affine hyperbolic plane was described by
8:
533:. In the case of the hyperbolic plane, such
180:, then its isotropy index is the minimum of
838:Introduction to Quadratic Forms over Fields
309:are isotropic. This example is called the
809:Quadratic forms chapter I: Witts theory
726:
395:⟨1⟩ ⊕ ⟨−1⟩
229:. If we consider the general element
7:
738:
736:
734:
732:
730:
827:Algebraic Theory of Quadratic Forms
445:the quadratic form is related to a
146:vectors in it are isotropic, and a
606:field, for example, the field of
154:(non-zero) isotropic vectors. The
25:
411:as a quadratic space with basis
269:are equivalent since there is a
860:Introduction to Quadratic Forms
553:A space with quadratic form is
1:
894:Graduate Texts in Mathematics
842:American Mathematical Society
285:, and vice versa. Evidently,
138:vector in it is isotropic, a
866:. p. 94 §42D Isotropy.
579:Witt's decomposition theorem
200:Not to be confused with the
245:, then the quadratic forms
91:) for that quadratic form.
954:
199:
140:totally isotropic subspace
746:; Husemoller, D. (1973).
647:Chevalley–Warning theorem
64:, then a non-zero vector
53:is a quadratic form on a
18:Metabolic quadratic space
748:Symmetric Bilinear Forms
715:Universal quadratic form
317:. A common instance has
447:symmetric bilinear form
150:if it does not contain
47:definite quadratic form
888:A Course in Arithmetic
190:pseudo-Euclidean space
49:. More explicitly, if
836:Tsit Yuen Lam (2005)
817:Coral Gables, Florida
587:orthogonal direct sum
563:orthogonal complement
549:Split quadratic space
543:hyperbolic-orthogonal
443:polarization identity
363:) = nonzero constant}
343:) = nonzero constant}
271:linear transformation
604:algebraically closed
399:bivariate polynomial
813:University of Miami
585:over a field is an
583:inner product space
206:hyperbolic geometry
882:Serre, Jean-Pierre
128:isotropic subspace
30:In mathematics, a
792:Geometric Algebra
710:Witt ring (forms)
369:. In particular,
313:in the theory of
148:definite subspace
16:(Redirected from
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798:Internet Archive
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754:. Vol. 73.
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311:hyperbolic plane
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196:Hyperbolic plane
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105:
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21:
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933:Quadratic forms
923:
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898:Springer-Verlag
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864:Springer-Verlag
854:
807:Pete L. Clark,
804:
803:
785:
781:
766:
756:Springer-Verlag
742:
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728:
723:
691:
673:
664:
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611:
608:complex numbers
595:
571:
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519:
471:
468:
465:
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462:
449:
424:
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404:are exhibited.
394:
393:. The notation
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315:quadratic forms
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108:quadratic space
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938:Bilinear forms
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831:W. A. Benjamin
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391:unit hyperbola
325:in which case
217:characteristic
215:be a field of
197:
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158:isotropy index
72:is said to be
41:is said to be
32:quadratic form
26:
24:
14:
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10:
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6:
4:
3:
2:
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849:0-8218-1095-2
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823:Tsit Yuen Lam
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818:
814:
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799:
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765:3-540-06009-X
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670:-adic numbers
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663:
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656:is the field
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137:
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126:is called an
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121:
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103:
99:
94:Suppose that
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71:
67:
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40:
37:
33:
19:
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859:
856:O'Meara, O.T
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631:finite field
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593:Field theory
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506:Two vectors
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441:Through the
440:
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384:
380:
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328:
323:real numbers
318:
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288:
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119:
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78:
73:
69:
65:
61:
57:
55:vector space
50:
42:
38:
29:
700:Polar space
423:satisfying
277:that makes
89:null vector
927:Categories
916:1034.11003
794:, page 119
787:Emil Artin
774:0292.10016
744:Milnor, J.
721:References
705:Witt group
516:orthogonal
409:Emil Artin
367:hyperbolas
281:look like
219:not 2 and
884:(2000) .
559:metabolic
167:signature
74:isotropic
43:isotropic
858:(1963).
689:See also
581:, every
379: :
355: :
335: :
116:subspace
825:(1973)
789:(1957)
475:
463:
389:is the
122:. Then
34:over a
914:
904:
870:
847:
772:
762:
610:, and
602:is an
387:) = 1}
811:from
629:is a
555:split
531:) = 0
518:when
432:= 0,
202:plane
114:is a
85:) = 0
60:over
36:field
902:ISBN
868:ISBN
845:ISBN
796:via
760:ISBN
672:and
633:and
557:(or
541:are
537:and
514:are
510:and
489:) −
461:) =
365:are
345:and
297:and
255:and
211:Let
184:and
136:some
110:and
912:Zbl
815:in
770:Zbl
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652:If
625:If
598:If
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273:on
241:of
204:in
152:any
144:all
142:if
134:if
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106:is
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68:in
929::
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503:.
501:))
497:−
485:+
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434:NM
428:=
421:}
417:,
375:∈
351:∈
331:∈
321:=
303:,
291:,
264:−
260:=
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250:=
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