Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, 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 (or null vector) for that quadratic form.

Suppose that (V, q) is

The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis {M, N} satisfying M² = N² = 0, NM = 1, where the products represent the quadratic form.^[2]

Through the polarization identity the quadratic form is related to a symmetric bilinear form B(u, v) = 1/4(q(u + v) − q(u − v)).

Two vectors u and v are

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.^[1]: 57 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.^{[1]: 12, 3}

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. By

• If F is an
  complex numbers
  , and (V, q) is a quadratic space of dimension at least two, then it is isotropic.
• If F is a finite field and (V, q) 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 Q_p of p-adic numbers and (V, q) is a quadratic space of dimension at least five, then it is isotropic.

See also

• Isotropic line
• Polar space
• Witt group
• Witt ring (forms)
• Universal quadratic form

References

• Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami in Coral Gables, Florida.
• W. A. Benjamin
  .
• Tsit Yuen Lam (2005) Introduction to Quadratic Forms over Fields,
  ISBN 0-8218-1095-2
  .
• ISBN 3-540-66564-1
  .
• Zbl 1034.11003
  .