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October 20, 2025Open Access

On the p-adic transcendence of ₊=₁^ p^-1/pᵏ

Key Points

The sum $ $ is a $p$-adic complex number, not algebraic, shedding light on transcendence.
It connects to Abhyankar's solution of the Artin-Schreier equation, expanding understanding of $p$-adic functions.
A conjecture is formulated regarding the order type of the support of algebraic $p$-adic Hahn series.
This conjecture is supported by an observation from Kedlaya, suggesting deeper implications within algebraic structures.

Abstract

Let p be a prime number. In this article, we prove that the p-adic Hahn series ₊=₁^ p^-1/pᵏ, which is the mixed-characteristic analogue of Abhyankar's solution ₊=₁^ t^-1/pᵏ to the Artin-Schreier equation Xᵖ-X-t^-1=0 over Fₚ (\! (t) \!), is a p-adic complex number, but not a p-adic algebraic number. Based on this result, we formulate a conjecture about the possible order type of the support of an algebraic p-adic Hahn series and prove that it is implied by a tentative observation of Kedlaya.

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Cite This Study

Wang et al. (Mon,) studied this question.

synapsesocial.com/papers/68f5fcce8d54a28a75cf1c96 https://doi.org/https://doi.org/10.48550/arxiv.2509.24609

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