Exercise 8, § 2

a) With the notation of exerc. 7, show that the series with general term $x^n / n!$ is convergent for every $x \in \mathfrak{p}$ if $p \neq 2$, for every $x \in \mathfrak{p}^2$ (but for no $x \notin \mathfrak{p}^2$) if $p = 2$ (determine the exponent of $p$ in the decomposition of $n!$ into prime factors). If $f(x)$ is the sum of this series show that $f$ is a continuous homomorphism of $\mathfrak{p}$ (resp. $\mathfrak{p}^2$) into $1 + \mathfrak{p}^2$. Deduce from this that for every $z \in \mathbf{Z}_p$ one has $f(pz) = e_p^z$ (resp. $f(p^2z) = e_p^{z'}$); in other words,

(use exerc. 7 e)).

b) For every $a \in 1 + \mathfrak{p}$ and every $x \in \mathbf{Z}_p$ show that $\log(a^x) = x \log a$, and deduce from a) and exerc. 7 d) that

c) For every $m \in \mathbf{Z}_p$ show that the continuous function $x \mapsto x^m$, defined on $1 + \mathfrak{p}$, admits a derivative equal to $mx^{m-1}$ (use b) and exerc. 7 e)).

d) Show that for every $m \in \mathbf{Z}_p$ and every $x \in \mathfrak{p}$ if $p \neq 2$ ($x \in \mathfrak{p}^2$ if $p = 2$) the series with general term $\binom{m}{n} x^n$ is convergent and that its sum is a continuous function of $m$; deduce from this that this sum is equal to $(1 + x)^m$ by remarking that $\mathbf{Z}$ is (everywhere) dense in $\mathbf{Z}_p$.