Common splitting fields of symbol algebras

Adam Chapman, Mathieu Florence, Kelly McKinnie

Research output: Contribution to journalArticlepeer-review

Abstract

We study the common splitting fields of symbol algebras of degree pm over fields F of char (F) = p. We first show that if any finite number of such algebras share a degree pm simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees pm0,⋯,pmt share a cyclic splitting field of degree pm0+⋯+mt. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol’s to bound the symbol length of classes in Brpm(F) whose symbol length when embedded into Brpm+1(F) is 2 for p∈ { 2 , 3 }. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

Original languageEnglish
Pages (from-to)649-662
Number of pages14
JournalManuscripta Mathematica
Volume171
Issue number3-4
DOIs
StatePublished - Jul 2023

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