Unramified cohomology of quadrics in characteristic two

Let F be a field of characteristic 2 and let X be a smooth projective quadric of dimension ≥ 1 over F. We study the unramified cohomology groups with 2-primary torsion coefficients of X in degrees 2 and 3. We determine completely the kernel and the cokernel of the natural map from the cohomology of F to the unramified cohomology of X. This extends the results in characteristic different from 2 obtained by Kahn, Rost and Sujatha in the nineteen-nineties.