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arXiv · 2111.07127

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

Shanwen Wang·Yijun Yuan
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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.

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Shanwen Wang, Yijun Yuan. 2021-11-13. Truncated expansion of $\zeta_{p^n}$ in the $p$-adic Mal'cev-Neumann field. https://arxiv.org/abs/2111.07127

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