Witt's Theorem - Witt's Decomposition Theorem

Witt's Decomposition Theorem

Let (V, q) be a quadratic space over a field k. Then it admits a Witt decomposition:

where V0=ker q is the radical of q, (Va, qa) is an anisotropic quadratic space and (Vh, qh) is a split quadratic space. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of (V, q) are determined uniquely up to isomorphism.

Quadratic forms with the same core form are said to be similar or Witt equivalent.

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