Isotropic quadratic form

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In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which it evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v)=0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector for that quadratic form.

Suppose that (V,q) is quadratic space and W is a subspace. Then W is called an isotropic subspace of V if all vectors in it are isotropic, and an anisotropic subspace if it does not contain any (non-zero) isotropic vectors. The isotropy index of a quadratic space is the maximum of the dimensions of the isotropic subspaces.

Contents 1 Examples 2 Relation with classification of quadratic forms 3 See also 4 References

[edit] Examples

1. A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form:

• either q is positive definite, i.e. q(v)>0 for all non-zero v in V ;
• or q is negative definite, i.e. q(v)<0 for all non-zero v in V.

More generally, if the quadratic form is non-degenerate and has the signature (p,q), then its isotropy index is the minimum of p and q.

2. If F is an algebraically closed field, for example, the field of complex numbers, and (V,q) is a quadratic space of dimension at least two, then it is isotropic.

3. If F is a finite field and (V,q) is a quadratic space of dimension at least three, then it is isotropic.

4. If F is the field Q_p of p-adic numbers and (V,q) is a quadratic space of dimension at least five, then it is isotropic.

[edit] Relation with classification of quadratic forms

From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field F, classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle.

[edit] See also

• Witt group
• Symmetric bilinear form

[edit] References

• Serre, Jean-Pierre, A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.
• Milnor, John and Dale Husemoller, Symmetric bilinear forms. Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. 1973.