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.