Difference between revisions of "Isotropic quadratic form"
From Encyclopedia of Mathematics
(Start article: Isotropic quadratic form) |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 3: | Line 3: | ||
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 $q(v) = 0$. | 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 | + | 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=35459
Isotropic quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropic_quadratic_form&oldid=35459