Bilinear Forms and Differential Forms under Field Extensions

15 June 2010
Detlev Hoffmann
An important problem in algebra is the study of algebraic objects defined over fields and how they behave under field extensions, for example the Brauer group of a field, Galois cohomology groups over fields, Milnor K-theory of a field, or the Witt ring of bilinear forms over a field. Of particular interest is the determination of the kernel of the restriction map when passing to a field extension. We will give an overview over some known results concerning the kernel of the restriction map from the Witt ring of a field to the Witt ring of an extension field. Over fields of characteristic not two, general results are rather sparse. In characteristic two, we have a much more complete picture. In this talk, I will explain the full solution to this problem for extensions that are given by function fields of hypersurfaces over fields of characteristic two. An important tool is the study of the behaviour of differential forms over fields of positive characteristic under field extensions. The result for Witt rings in characteristic two then follows by applying earlier results by Kato, Aravire-Baeza, and Laghribi. This is joint work with Andrew Dolphin.