Abstract: Let $F$ be a field of characteristic $p>0$, and let $X$ be an integral affine hypersurface over $F$ with function field $K=F(X)$. We determine the kernels of the restriction maps given by extending scalars from $F$ to $K$ for the following algebraic objects: the space of absolute Kahler differentials, Milnor $K$-theory modulo $p$, and the Witt ring of symmetric bilinear forms in the case $p=2$. All these kernels have a surprisingly explicit description. This is joint work by Andrew Dolphin and myself, the case of Milnor $K$-theory being due to Stephen Scully who uses our results on differential forms.