Is $e^p\in\mathbb{Q}_p$ known to be transcendental? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:55:40Z http://mathoverflow.net/feeds/question/111387 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111387/is-ep-in-mathbbq-p-known-to-be-transcendental Is $e^p\in\mathbb{Q}_p$ known to be transcendental? 36min 2012-11-03T16:12:41Z 2012-11-03T21:12:20Z <p>$\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}$ doesn't converge in <code>$\mathbb{Q}_p$</code>, however, $e^p:=\sum\limits_{n=0}^{\infty}\dfrac{p^n}{n!}$ does converge for <code>$p\neq 2$</code>. So my question is,</p> <blockquote> <p>Are $e^p\in\mathbb{Q}_p$ for $p\neq2$ (and $e^4\in\mathbb{Q}_2$) known to be non-algebraic numbers?</p> </blockquote> http://mathoverflow.net/questions/111387/is-ep-in-mathbbq-p-known-to-be-transcendental/111390#111390 Answer by René Pannekoek for Is $e^p\in\mathbb{Q}_p$ known to be transcendental? René Pannekoek 2012-11-03T16:25:11Z 2012-11-03T21:12:20Z <p>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):</p> <p><a href="http://www.jstor.org/discover/10.2307/2373193?uid=3738736&amp;uid=2&amp;uid=4&amp;sid=21101389828747" rel="nofollow">http://www.jstor.org/discover/10.2307/2373193?uid=3738736&amp;uid=2&amp;uid=4&amp;sid=21101389828747</a></p> <p>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 $\left| a \right| &lt; p^{-1/(p-1)}$, then $\exp(a) \in \mathbf{Q}_p$ is transcendental over $\mathbf{Q}$ (ibid.).</p>