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

TL;DR

This paper derives a truncated $p$-adic expansion of roots of cyclotomic polynomials using a transfinite Newton algorithm, revealing new harmonic number identities that simplify the expansion expression.

Contribution

It introduces a novel truncated expansion of $oldsymbol{ ext{ extit{p}}}$-adic roots of cyclotomic polynomials using a transfinite Newton method, along with a new harmonic number identity.

Findings

Provides a $rac{2}{(p-1)p^{n-2}}$-truncated expansion of $oldsymbol{ ext{ extit{zeta}}_{p^n}}$.

Establishes a new harmonic number identity relevant to $p$-adic expansions.

First to explicitly describe the first $oldsymbol{eth_0^2}$ terms of this expansion.

Abstract

Fix an odd prime $p$. In this article, we provide a $\mathrm{mod}\ p$ harmonic number identity, which appears naturally in the canonical expansion of a root $\zeta_{p^n}$ of the $p^n$-th cyclotomic polynomial $\Phi_{p^n}(T)$ 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 $\aleph_0^2$ terms of the canonical expansion of $\zeta_{p^n}$. The harmonic number identity simplifies the expression of this expansion.