Common splitting fields of symbol algebras

Abstract

We study the common splitting fields of symbol algebras of degree \(p^m\) over fields F of \({\text {char}}(F)=p\). We first show that if any finite number of such algebras share a degree \(p^m\) 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 \(p^{m_0},\dots ,p^{m_t}\) share a cyclic splitting field of degree \(p^{m_0+\dots +m_t}\). 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 \({\text {Br}}_{p^m}(F)\) whose symbol length when embedded into \({\text {Br}}_{p^{m+1}}(F)\) is 2 for \(p\in \{2,3\}\). We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.