Linkage of Symbol $p$-Algebras of Degree 3

TL;DR

This paper investigates conditions under which certain degree 3 symbol p-algebras over fields of characteristic 3 share common splitting fields, linking cohomological triviality to algebraic extensions.

Contribution

It establishes a connection between Kato-Milne cohomology triviality and the existence of inseparable degree 3 splitting fields for these algebras.

Findings

If the cohomology class is trivial, the algebras share an inseparable degree 3 splitting field.

Over quadratically closed fields, trivial cohomology implies a shared inseparable degree 3 extension.

The results relate cohomological conditions to explicit algebraic extensions in characteristic 3.

Abstract

Given a field $F$ of characteristic $3$ and division symbol $p$-algebras $[\alpha,\beta)_{3,F}$ and $[\alpha,\gamma)_{3,F}$ of degree $3$ over $F$, we prove that if $\alpha \text{dlog}(\beta)\wedge \text{dlog}(\gamma)$ is trivial in the Kato-Milne cohomology group $H_3^3(F)$ then the algebras share a common splitting field which is an inseparable degree 3 extension of either $F$ or a quadratic extension of $F$. In the special case of quadratically closed fields, if $\alpha \text{dlog}(\beta)\wedge \text{dlog}(\gamma)=0$, then they share an inseparable degree 3 extension of $F$.