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Difference between revisions of "Isotropic quadratic form"

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A [[quadratic form]] on a [[vector space]] over a field F which is non-degenerate (the associated [[bilinear form]] is non-singular) but which represents zero non-trivially: there is a non-zero vector v such that q(v) = 0.
 
A [[quadratic form]] q on a [[vector space]] over a field F which is non-degenerate (the associated [[bilinear form]] is non-singular) but which represents zero non-trivially: there is a non-zero vector v such that q(v) = 0.
  
An '''anisotropic quadratic form'''' q is one for which  q(v) = 0 \Rightarrow v=0.
+
An '''anisotropic quadratic form''' q is one for which  q(v) = 0 \Rightarrow v=0.
  
 
====References====
 
====References====
 
* Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'',  Graduate Studies in Mathematics '''67''',  American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZBL|1068.11023}} {{MR|2104929 }}  
 
* Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'',  Graduate Studies in Mathematics '''67''',  American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZBL|1068.11023}} {{MR|2104929 }}  
 
* J.W. Milnor, D. Husemöller, ''Symmetric bilinear forms'', Ergebnisse der Mathematik und ihrer Grenzgebiete '''73''', Springer-Verlag (1973) {{ISBN|0-387-06009-X}} {{ZBL|0292.10016}}
 
* J.W. Milnor, D. Husemöller, ''Symmetric bilinear forms'', Ergebnisse der Mathematik und ihrer Grenzgebiete '''73''', Springer-Verlag (1973) {{ISBN|0-387-06009-X}} {{ZBL|0292.10016}}

Latest revision as of 19:32, 15 November 2023

2020 Mathematics Subject Classification: Primary: 15A63 [MSN][ZBL]

A quadratic form q on a vector space over a field F which is non-degenerate (the associated bilinear form is non-singular) but which represents zero non-trivially: there is a non-zero vector v such that q(v) = 0.

An anisotropic quadratic form q is one for which q(v) = 0 \Rightarrow v=0.

References

  • Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
  • J.W. Milnor, D. Husemöller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag (1973) ISBN 0-387-06009-X Zbl 0292.10016
How to Cite This Entry:
Isotropic quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropic_quadratic_form&oldid=54476