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Central simple algebras of prime exponent and divided power operations

Published online by Cambridge University Press:  15 January 2013

A.S. Sivatski
A.S. Sivatski*
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Brasilalexander.sivatski@gmail.com
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Abstract

Let 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.

Keywords

Cyclic algebrasMilnor K-groupsDivided power operationsNorm residue isomorphism16K5019D45
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 113 - 123
Copyright
Copyright © ISOPP 2013

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