Common Splitting Fields of Symbol Algebras

We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. 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 $\operatorname{Br}_{p^m}(F)$ whose symbol length when embedded into $\operatorname{Br}_{p^{m+1}}(F)$ 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.