Consider the $p$-adic exponential defined over $\mathbb C\_p$. One knows $\exp$ is analytic in the domain $\mathcal D=\{z\in\mathbb C\_P\mid v\_p(z)>\frac1{p-1}\}$. Does it exist an element $z\_0\in\mathcal D$ such that $\exp(z\_0)=0$?
Thanks in advance.



Zero of the exponential p-adic
The exponential function satisfies $\exp \left({x + y}\right) = \exp\left(x \right) \exp\left(y \right)$ for $x, y$ in the convergence domain. It also satisfies $\exp \left( 0 \right) = 1$. So if $\exp \left( z\_0 \right) = 0$, then $0 = \exp \left( z\_0 \right) \cdot \exp\left(-z\_0 \right) = \exp \left( z\_0 + (-z\_0) \right) = \exp\left( 0 \right) = 1$, which is a contradiction.


