Truncated expansion of ${\zeta }_{p^n}$ in the $p$-adic Mal'cev-Neumann field

Fix an odd prime $p$. In this article, we provide a $\mathrm{m}\mathrm{o}\mathrm{d}p$ harmonic number identity, which appears naturally in the canonical expansion of a root ${\zeta }_{p^n}$ of the $p^n$-th cyclotomic polynomial ${\mathrm{\Phi }}_{p^n}(T)$ in the $p$-adic Mal’cev-Neumann field ${\mathbb{L}}_p$. We establish a $\frac{2}{(p-1)p^{n-2}}$-truncated expansion of ${\zeta }_{p^n}$ via a variant of the transfinite Newton algorithm, which gives the first ${\mathrm{\aleph }}_0^2$ terms of the canonical expansion of ${\zeta }_{p^n}$. The harmonic number identity simplifies the expression of this expansion.