Characterization of Hermitian trace forms
Let \(q\) be a nondegenerate quadratic form in the even number \(m\) of variables over the field \(K\) of characteristic different from 2. \textit{M. Epkenhans} [Arch. Math. 60, 527-529 (1993; Zbl 0777.11009)] has given necessary and sufficient conditions for \(q\) to be of the form \(q(x) = \text{Tr}_{E/K}(x^2)\) for a separable extension \(E\) of \(K\) of degree \(m\). The author modifies this question and asks when is \(q\) of the form \(q(x) = \text{Tr}_{E/K}(x x^{\sigma})\) for an involution \(\sigma\) of \(E/K\) as above. He shows that this is the case if and only if the total signature of \(q\) is nonnegative and either \(m \geq 4\) or \(m = 2\) and \(q \cong \langle 2,-2D\rangle\) holds with \(D \in K^{\times}- (K^{\times})^2\).
scientific article; zbMATH DE number 1251416