Witt kernels and Brauer kernels for quartic extensions in characteristic two

Let F   be a field of characteristic 2 and let E/FE/F be a field extension of degree 4. We determine the kernel Wq(E/F)Wq(E/F) of the restriction map WqF→WqEWqF→WqE between the Witt groups of nondegenerate quadratic forms over F and over E  , completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduce the corresponding result for the Witt kernel W(E/F)W(E/F) of the restriction map WF→WEWF→WE between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As an application, we describe the 2-torsion part of the Brauer kernel for such extensions.