WittIndex -- returns the Witt index of a symmetric bilinear form

Synopsis

• Usage:

  WittIndex(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, denoted by $\beta\in\text{GW}(k)$, where $k$ is the complex numbers, reals, rationals, or a finite field
• Outputs:
  • an integer, the rank of the totally isotropic part of $\beta$

Description

By Witt decomposition, any form decomposes uniquely as $\beta \cong k \mathbb{H} \oplus \beta_a$ where the form $\beta_a$ is anisotropic. The integer $k$ is called the Witt index of $\beta$. See for instance [L05, I.4.3].

Citations:

• [L05] T.Y. Lam, Introduction to quadratic forms over fields, American Mathematical Society, 2005.

See also

• anisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form
• anisotropicDimensionQp -- returns the anisotropic dimension of a rational symmetric bilinear form over the p-adics