The orthogonal \(u\)-invariant of a quaternion algebra
In particular, the following two theorems are proved: Theorem 4.10: Let \(Q\) be a quaternion division algebra over \(K\) with canonical involution \(\gamma\). Then the set \(\text{Herm}^{-1}_1(Q,\gamma)\) of (equivalence classes of) one-dimensional skew-Hermitian forms on \((Q,\gamma)\) is finite iff \(K^x/K^{x^2}\) is finite. Theorem 6.2/6.8: Let \(K\) be a Kaplansky field with its unique quaternion division algebra \(Q\). Then any skew-Hermitian form of dimension \(\geq 4\) over \((Q,\gamma)\) is isotropic. The corresponding \(u\)-invariant \(u(Q,\gamma,-1)\) has the value \[ \begin{cases} 1 \quad\text{if }K \text{ is real Euclidean}\\ 2 \quad\text{if }K \text{ is real non-Euclidean} \\ 3 \quad\text{if }K \text{ is nonreal}.\end{cases} \] (Note that \(\gamma\) is a symplectic involution). This paper is well-written and easy to read.
scientific article; zbMATH DE number 5689259