Cohomology and graded Witt group kernels for extensions of degree four in characteristic two

In earlier work the authors determined the graded Witt kernel GWq(E/F)=ker(GWqF→GWqE)GWq(E/F)=ker(GWqF→GWqE) when E/FE/F is a biquadratic extension in characteristic 2 by calculating the cohomological kernel H2⁎(E/F)=ker(H2⁎F→H2⁎E). In this paper this result is extended to the cases where [E:F]=4[E:F]=4 and E is either cyclic or has dihedral Galois closure. In addition, the use of Izhboldin's Q-groups is generalized to obtain six-term exact sequences that describe the behavior of these graded rings whenever four-term exact sequences and homotopies describe the arithmetic of the extension. These tools are valid in characteristic p, although the applications here are in characteristic 2.