In mathematics, a quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

 over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 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

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 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

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

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

Examples

1. The hyperbolic plane is a two-dimensional isotropic quadratic space with the form xy.

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

Definite bilinear form

In mathematics, a definite bilinear form is a bilinear form B over some vector space V such that the associated quadratic formQ=B \,...

:

• 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.

3. 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.

4. If F is a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 and (V,q) is a quadratic space of dimension at least three, then it is isotropic.

5. If F is the field Q_p of p-adic number

P-adic number

In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s and (V,q) is a quadratic space of dimension at least five, then it is isotropic.

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.