The de Rham Witt complex, cohomological kernels and pm-extensions in characteristic p

In earlier work the authors determined the Brauer kernel of extensions of degree p in characteristic p>2 where the Galois group is a semidirect product of order ps for s|(p−1). This result is extended here and tools are developed to compute the cohomological kernels Hpmn+1(Em/F) for all n≥0 where [Em:F]=pm and the Galois closure is a semidirect product of cyclic groups order pm and s where s|(p−1). A six-term exact sequence describing the K-theory and cohomology of the extension is obtained. As an application it is shown that any F-division p-algebra of index pm split in Em is cyclic; a characteristic p analogue of a result of Vishne. The proofs use the de Rham Witt complex and Izhboldin groups, extending techniques developed earlier for the study of degree 4 extensions in characteristic two. The paper also provides background on the de Rham Witt Complex and Izhboldin groups difficult to track down in the literature.