Central simple algebras of prime exponent and divided power operations

Abstract

AbstractLet p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.