Skew-Hermitian forms which become hyperbolic over a splitting field (Q2708181)

Let \(H\) be a quaternion division algebra with canonical involution \(\tau\) over a field \(F\) of characteristic \(\neq 2\). Let \(K\) be a generic splitting field over \(F\) (for example, the function field of the Severi-Brauer variety associated to \(H\)). Using scalar extension and Morita equivalence, one obtains a map \(\rho\) mapping skew-Hermitian forms over \((H,\tau)\) to quadratic forms over \(K\), thus inducing a homomorphism of the respective Witt groups \(W^{-1}(H,\tau)\to W(K)\). By invoking some well-known exact sequences of Witt groups, it is shown that this homomorphism is injective. In fact, as a corollary, using a result by Parimala, Sridharan and Suresh, it is shown that \(\rho\) maps anisotropic skew-Hermitian forms onto anisotropic quadratic forms.