Some split symbol algebras of prime degree

Let $p$ be an odd prime, let $K=\mathbb{Q}(ϵ)$ where $ϵ$ is a primitive cubic root of unity, and let $L$ be the Kummer field $\mathbb{Q}(ϵ,\sqrt[3]{\alpha })$. In this paper we obtain a characterization of the splitting behavior of the symbol algebras $\left( \frac{\alpha ,p}{K,\epsilon }\right)$and$\left( \frac{\alpha ,p^{h_{p}}}{K,\epsilon}\right)$,where$h_{p}$istheorderintheclassgroup$Cl\left(L\right)$ofaprimeidealof$$\mathcal{O}$_L$whichdivides$p$\mathcal{O}$_L.$