Unramified cohomology of quadrics, I

Given a quadric X over a field F of characteristic ≠ 2, we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 Galois cohomology of F to the unramified mod 2 cohomology of F(X), when dim X > 10 and in several smaller-dimensional cases. Applications of these results to real quadrics and to the unramified Witt ring are given.