Is $e^p\in\mathbb{Q}_p$ known to be transcendental?

2012-11-03T16:12:41Z

$\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}$ doesn't converge in $\mathbb{Q}_p$ , however, $e^p:=\sum\limits_{n=0}^{\infty}\dfrac{p^n}{n!}$ does converge for $p\neq 2$ . So my question is,

Are $e^p\in\mathbb{Q}_p$ for $p\neq2$ (and $e^4\in\mathbb{Q}_2$) known to be non-algebraic numbers?

Answer by René Pannekoek for Is $e^p\in\mathbb{Q}_p$ known to be transcendental?

2012-11-03T16:25:11Z

According to the last paragraph in Section 3 of the paper "Transcendental numbers in the p-adic domain" by William W. Adams (Amer. J. of Math., Vol. 88, 1966):

http://www.jstor.org/discover/10.2307/2373193?uid=3738736&uid=2&uid=4&sid=21101389828747

the answer is yes. More specifically, they prove that if $a \in \mathbf{Q}_p$ is a non-zero element, algebraic over $\mathbf{Q}$, such that $\left| a \right| < p^{-1/(p-1)}$, then $\exp(a) \in \mathbf{Q}_p$ is transcendental over $\mathbf{Q}$ (ibid.).

Is $e^p\in\mathbb{Q}_p$ known to be transcendental? - MathOverflow

Is $e^p\in\mathbb{Q}_p$ known to be transcendental?

Answer by René Pannekoek for Is $e^p\in\mathbb{Q}_p$ known to be transcendental?