Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article we classify the groups that can arise as $E(\mathbb{Q}(\zeta_p))_{\text{tors}}$ up to isomorphism, where $p$ is a prime. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of $E(\mathbb{Q}^{ab})_{\text{tors}}.$