Common Splitting Fields of Symbol Algebras

TL;DR

This paper investigates the shared splitting fields of symbol algebras over fields of characteristic p, establishing conditions under which multiple algebras share cyclic splitting fields and exploring implications for the structure of the Brauer group.

Contribution

It proves that multiple symbol algebras sharing a purely inseparable splitting field also share a cyclic splitting field, generalizing known tensor product results and bounding symbol lengths in Brauer groups.

Findings

Shared splitting fields imply cyclic splitting fields of combined degree.

Tensor products of symbol algebras are themselves symbol algebras.

Bounded symbol length in Brauer groups for certain classes.

Abstract

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.