Let $n\geq 3$, and let $p$ be a prime number $\equiv 1 $ mod $n$.
In complex numbers, we can write a primitive $n$-th root of unity as $\exp(2\pi i/n)$.
Also, by Hensel's lemma, we see that $n$-th cyclotomic field is contained in $\mathbb{Q}_p$.
Indeed, the roots of $X^n -1 = 0$ are contained in $\mathbb{Z}_p$. Denote by $\eta_n$ a primitive $n$-th root of unity in $\mathbb{Z}_p$.
Can we use the complex number $\exp(2\pi i/n)$ to figure out the expansion of $\eta_n$ in $p$-adic integers?