On the $p$-adic transcendence of $\sum_{k=1}^\infty p^{-1/p^k}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915521773961216 |
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| author | Wang, Shanwen Yuan, Yijun |
| author_facet | Wang, Shanwen Yuan, Yijun |
| contents | Let $p$ be a prime number. In this article, we prove that the $p$-adic Hahn series $\sum_{k=1}^\infty p^{-1/p^k}$, which is the mixed-characteristic analogue of Abhyankar's solution $\sum_{k=1}^\infty t^{-1/p^k}$ to the Artin-Schreier equation $X^p-X-t^{-1}=0$ over $\mathbf{F}_p\left(\!\left(t\right)\!\right)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $p$-adic transcendence of $\sum_{k=1}^\infty p^{-1/p^k}$ Wang, Shanwen Yuan, Yijun Number Theory 11J81, 11J61, 11D88 Let $p$ be a prime number. In this article, we prove that the $p$-adic Hahn series $\sum_{k=1}^\infty p^{-1/p^k}$, which is the mixed-characteristic analogue of Abhyankar's solution $\sum_{k=1}^\infty t^{-1/p^k}$ to the Artin-Schreier equation $X^p-X-t^{-1}=0$ over $\mathbf{F}_p\left(\!\left(t\right)\!\right)$, 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. |
| title | On the $p$-adic transcendence of $\sum_{k=1}^\infty p^{-1/p^k}$ |
| topic | Number Theory 11J81, 11J61, 11D88 |
| url | https://arxiv.org/abs/2509.24609 |