$p$-adic exponential and Strassmans theorem

Question

I want to show that the $p$-adic exponential function is not periodic (or, equivalently, that there is no $x \in \mathbb{Z}_p, x \neq 0$ with $\exp(x)=1$). I read that this can be showed with Strassmans theorem, but I am not sure why (and how) this theorem may be applied to the series defining $\exp$. Can anyone help me?

Answer 1

It’s pretty well known that the exponential series $\exp(x)$ is convergent only for $v_p(x)>1/(p-1)$. If you do take $x$ in that range, $v_p\bigl(\exp(x)-1\bigr)=v_p(x)$, because the first term in the series for $\exp(x)-1$ dominates. Thus no possibility that the value of the exponential can be $1$ for nonzero $x$.

I’m not aware of any extension of the exponential beyond the domain of convergence of the series.