Eva Bayer-Fluckiger

FORMS IN ODD DEGREE EXTENSIONS AND SELF-DUAL NORMAL BASES

Introduction. Let K be a field. Springer has proved that an anisotropic quadratic form over K is also anisotropic over any odd degree extension of K (see [31], [14]). If the characteristic of K is not 2, this implies that two nonsingular quadratic forms that become isomorphic over an extension of odd degree of K are already isomorphic over K (see [31]). In [27], Serre reformulated the latter Statement äs follows: if O is an orthogonal group over K, then the canonical map of Galois cohomology sets

Hermitian categories, extension of scalars and systems of sesquilinear forms

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e., the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends an earlier result of the first and third authors. We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category and to generalize results such as Witts cancellation theorem, Springers theorem, the weak Hasse principle, and finiteness of genus to systems of sesquilinear forms over hermitian categories.

Hasse principle for G-trace forms

Let be a global field of characteristic not 2. We prove a local-global principle for the existence of self-dual normal bases, and more generally for the isomorphism of -trace forms, for -Galois algebras over .

Embeddings of maximal tori in classical groups over local and global fields

Embeddings of maximal tori in classical groups over fields of characteristic not 2 are the subject matter of several recent papers. The aim of the present paper is to give necessary and sufficient conditions for such an embedding to exist, when the base field is a local field, or the field of real numbers. This completes the results of [3], where a complete criterion is given for the Hasse principle to hold when the base field is a global field.

Multiples of forms

The aim of this paper is to survey and extend some results concerning multiples of (quadratic, hermitian, bilinear…) forms.