Isotropic quadratic form

s Let F be a field of characteristic not 2 and . If we consider the general element of V, then the quadratic forms and are equivalent since there is a linear transformation on V that makes q look like r, and vice versa. Evidently, and are isotropic. This example is called the hyperbolic plane in the theory of quadratic forms. A common instance has F = real numbers in which case {{nowrap|1={x ∈ V : q(x) = nonzero constant} }} and {{nowrap|1={x ∈ V : r(x) = nonzero constant} }} are hyperbolas. In particular, {{nowrap|1={x ∈ V : r(x) = 1} }} is the unit hyperbola. The notation has been used by Milnor and Husemoller Through the polarization identity the quadratic form is related to a symmetric bilinear form . Two vectors u and v are orthogonal when . In the case of the hyperbolic plane, such u and v are hyperbolic-orthogonal. ==Split quadratic space==

A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; 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. == Relation with classification of quadratic forms ==

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 F, classification of definite quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By Witt's decomposition theorem, every inner product space over a field is an orthogonal direct sum of a split space and a space with definite quadratic form. ==Field theory==

• If F is an algebraically closed field, for example, the field of complex numbers, and is a quadratic space of dimension at least two, then it is isotropic. • If F is a finite field and is a quadratic space of dimension at least three, then it is isotropic (this is a consequence of the Chevalley–Warning theorem). • If F is the field Qp of p-adic numbers and is a quadratic space of dimension at least five, then it is isotropic. == See also ==