On hermitian trace forms over hilbertian fields

摘要

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 $muin E^*$ , we define a hermitian scaled trace form by $xin Emapstomathrm{Tr}_{E/k}(mu x x^sigma)$ . 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.