On hermitian trace forms over hilbertian fields

Let k be a field of characteristic different from 2. Let E/k be a finite separable extension with a k-linear involution σ. For every σ-symmetric element μ ∈ E∗, we define a hermitian scaled trace form by x ∈ E 7→ TrE/k(μxx). If μ = 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. Introduction: In this paper, the words “quadratic form” are reserved to mean “non-degenerate quadratic form”. Let k be a field of characteristic different from 2. If E/k is a finite separable extension and λ ∈ k∗, we can define a quadratic form E → k, x 7→ TrE/k(λx), denoted by TrE/k(< λ >). Such a form is called a scaled trace form. If λ = 1 this form is called the trace form of E/k. Recall that a field k is hilbertian if Hilbert’s irreducibility theorem holds. To simplify, k is hilbertian if for all n,m ≥ 1 and for all irreducible polynomial P ∈ k(Y1, · · · , Ym)[X1, · · · , Xn], there exist infinitely many specializations (y1, · · · , ym) ∈ k such that P (y1, · · · , ym, X1, · · · , Xn) is still irreducible (for a precise statement of this theorem, see [La] for example). A natural problem is to know which quadratic forms over k are isomorphic, or more reasonably Witt-equivalent, to a (scaled) trace form. No answer has been given in general, but Scharlau and Waterhouse gave independently a characterization of scaled trace forms over a hilbertian field (see Theorem 1 below). The characterization of trace forms, which is much more difficult, has been initiated by Conner and Perlis, who were interested in the following question: which quadratic forms over Q are Witt-equivalent to a trace form ? In [C-P], they showed that such forms are precisely the positive quadratic forms (Recall that a quadratic form over a field k is called positive if all its signatures are non-negative). In [S], Scharlau showed that the result is always true when k is a number field. Finally, Krüskemper and Scharlau proved the validity of this result for some hilbertian fields (see Theorem 3 below). Note that a characterization of isometry classes of trace forms has been obtained by Epkenhans in [E] when k is a number field.