Linkage of Symbol $p$-Algebras of Degree 3

Abstract

Given a field $F$ of characteristic $3$ and division symbol $p$-algebras $[\hat{I}\pm ,{\hat{I}}^2{)}_{3,F}$ and $[\hat{I}\pm ,{\hat{I}}^3{)}_{3,F}$ of degree $3$ over $F$, we prove that if $\hat{I}\pm \text{dlog}({\hat{I}}^2)\wedge \text{dlog}({\hat{I}}^3)$ 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 $\hat{I}\pm \text{dlog}({\hat{I}}^2)\wedge \text{dlog}({\hat{I}}^3)=0$, then they share an inseparable degree 3 extension of $F$.