On the $p$-adic transcendence of ${\sum }_{k=1}^{\infty }{p}^{-1/p^k}$

Abstract

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.

10 pages