Trace forms of central simple algebras over a local field or a global field

Introduction: Let A be a central simple algebra over a field k of characteristic different from 2. The quadratic form x ∈ A 7→ TrdA(x ) ∈ k is called the trace form of A, and is denoted by TA. This trace form has been studied by many authors (cf.[L], [LM], [Ti] et [Se], Annexe §5 for example). In particular, these classical invariants are well-known (loc.cit.). In [B], we have determined a diagonalization of the trace form of cyclic algebras over fields such that the cube of the fundamental ideal is trivial. In particular, this gives all trace forms of central simple algebras over local fields and non formally real global fields, since any algebra over such a field is cyclic. In this article, we give a characterization of the trace form of a central simple algebra over a local field or a global field in terms of determinant and signatures. We also show that a quadratic form over a global field is isomorphic to a trace form if and only if it is true locally. Then we give a necessary and sufficient condition on trace forms to be isomorphic. Finally, we apply these results to describe explicitly the elements of Br2(k) which can be written n 2 [A], where A is a