Some split symbol algebras of prime degree

Abstract

Let $p$ be an odd prime, let $K=\mathbb{Q}(\epsilon )$ where $\epsilon$ is a primitive cubic root of unity, and let $L$ be the Kummer field $\mathbb{Q}\left(\epsilon ,\sqrt[3]{\alpha }\right)$. 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 $h_p$ is the order in the class group $Cl\left(L\right)$ of a prime ideal of ${\mathcal{O}}_L$ which divides $p{\mathcal{O}}_L.$