  Let $p\geq 3$ be a prime number. In this article, we study the canonical
expansion of the primitive $p^n$-th root of unity $\zeta_{p^n}$ in $p$-adic
Mal'cev-Neumann field $\mathbb{L}_p$ for $n\geq 1$. More precisely, we give the
explicit formula for the first $\aleph_0$ terms of the expansion of
$\zeta_{p^n}$ and as an application, we use it to construct a uniformizer of
$K_{2,m}=\mathbb{Q}_p\left(\zeta_{p^2},p^{1/p^m}\right)$ with $m\geq 1$.
