Let F be a field of characteristic not 2 and X be a quadric over F. In this
paper, we study the kernel and cokernel of the natural map from the
Galois cohomology of F, more precisely H^i(F,Q/Z(i-1)), to the
corresponding unramified cohomology of the function field of X. In
particular we show that, for i=4, the kernel has order at most 2 if dim X
> 6 and is 0 if dim X > 14, and the cokernel has order at most 4 if dim X
is not 4 and is 0 if dim X > 10. These results have applications to the
unramified Witt ring of F(X).
A more recent version is here.