LAUSR

Abstract 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 H p m n + 1 ( E m / F ) for all n ≥ 0 where [ E m : F ] = p m and the Galois closure is a semidirect product of cyclic groups order p m 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 p m split in E m 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.