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.