TITLE: irrationality of the p-adic exponential
QUESTION [7 upvotes]: I would like to illustrate my lecture on p-adic numbers with some elementary results.
I proved that the series $e^p=\sum_{n\ge0}\frac{p^n}{n!}$ converges in $\mathbb Q_p$ for every prime $p$.
Now I would like to teach that $e^p$ is irrational in $\mathbb Q_p$ by elementary methods (this is true by Mahler result on transcendence of $e^x$ in $p$-adic fields).
Do you know such an elementary proof. I can't find one in the literature...
Good catch! I meant $\frac{p^n}{n!}$ instead of $\frac1n$. I edited it.

REPLY [2 votes]: Maybe I harp on this too much, but I think the $p$-adic exponential is not nearly as interesting as the logarithm. It’s defined on a much larger subgroup than the exponential, it’s a homomorphism (unlike the complex logarithm), and its roots are the $p$-power roots of unity. As a function on $1+D$, where $D$ is the open unit disk in $\Bbb C_p$, it’s onto $\Bbb C_p$, so that even though for $z\in\Bbb C_p$ you won’t usually be able to define $e^z$, you can find $\alpha\in\Bbb C_p^*$ such that $\log\alpha=z$.