| The cohomology groups associated to the absolute Galois group of a field E encode a great deal of information about E, with the groups Hm(GE, mu p) being of classical interest. These groups are linked to the reduced Milnor K-groups KmE/pK mE = kmE by the Bloch-Kato conjecture. Using this conjecture when E/F is a Galois extension of fields with Gal(E/F) ≃ Z/pnZ for some odd prime p, and additionally assuming xi p ∈ E, we study the groups H m(GE, mup) as modules over the group ring Fp [Gal(E/F)]. When E/F embeds in an extension E'/F with Gal(E'/F) ≃ Z/pn+1 Z , we are able to give a highly stratified decomposition of Hm(GE, mup). This allows us to give a decomposition of the cohomology groups of a p-adic extension of fields. In general we are able to give a coarse decomposition of Hm(GE, mu p), showing that many indecomposable types do not appear in Hm(GE, mup). With an additional assumption about the norm map Nnn-1 : kmEn → k mEn-1, we strengthen this coarse decomposition to a highly stratified one. |
