Proposition 13.9. Assuming that:

$r > \frac{e}{p - 1}$

Then

\exp (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

converges on

π^{r} O_{K}

and induces an isomorphism

(π^{r} O_{K}, +) \tilde{\to} (1 + π^{r} O_{K}, \times) .