On hermitian trace forms over hilbertian fields

Abstract

Let k be a field of characteristic different from 2. Let \(E/k\) be a finite separable extension with a {\it k}-linear involution \(\sigma\). For every \(\sigma\)-symmetric element \(\mu\in E^*\), we define a hermitian scaled trace form by \(x\in E\mapsto\mathrm{Tr}_{E/k}(\mu x x^\sigma)\). If \(\mu=1\), it is called a hermitian trace form. In the following, we show that every even-dimensional quadratic form over a hilbertian field, which is not isomorphic to the hyperbolic plane, is isomorphic to a hermitian scaled trace form. Then we give a characterization of Witt classes of hermitian trace forms over some hilbertian fields.