Let $p \in \mathbb{Z}$ be a prime and consider the number field $k = \mathbb{Q}[x]/(x^{(p^2 -1)/2} - p)$. We shall denote by $O_k$ the ring of integers of $k$. Let $\beta \in O_k$ be such that $\beta^{(p^2-1)/2} = p$. Then it is clear that $(\beta)$ is a prime ideal of $O_k$. Let $\widehat{O}_k$ be the completion of $O_k$ with respect to $(\beta)$.

My question is what are the roots of unity in $\widehat{O}_k$, in particular, does it have the $p+1$ -st roots of unity?

PS. Since $O_k/(\beta) = \mathbb{F}_p$ it is easy to see that $\widehat{O}_k$ has all the $p-1$ -st roots of unity using Hensel's lemma, but I could not infer anything about the $p + 1$ -st roots.