Exercise 7, § 2

Let $p$ be a prime number and $\mathbf{Q}_p$ the field of $p$-adic numbers (Gen. Top., III, p. 322 and 323, exercs. 23 to 25), let $\mathbf{Z}_p$ be the ring of $p$-adic integers, and $\mathfrak{p}$ the principal ideal $(p)$ in $\mathbf{Z}_p$.

a) Let $a = 1 + pb$, where $b \in \mathbf{Z}_p$ is an element of the multiplicative group $1 + \mathfrak{p}$; show that when the rational integer $m$ increases indefinitely the $p$-adic number $\frac{(1 + pb)^{p^m} - 1}{p^m}$ tends to a limit equal to the sum of the convergent series

One denotes this limit by $\log a$.

b) Show that when the $p$-adic number $x$ tends to 0 in $\mathbf{Q}_p$ the number $\frac{a^x - 1}{x}$ tends to $\log a$ (use $a$) and the definition of the topology of $\mathbf{Q}_p$.

c) Show that if $p \neq 2$ one has $\log a \equiv pb \pmod{\mathfrak{p}^2}$, and if $p = 2$ then $\log a \equiv 0 \pmod{\mathfrak{p}^2}$, and $\log a \equiv -4b^4 \pmod{\mathfrak{p}^3}$.

d) Show that if $p \neq 2$ (resp. $p = 2$) then $x \mapsto \log x$ is an isomorphism of the multiplicative topological group $1 + \mathfrak{p}$ onto the additive topological group $\mathfrak{p}$ (resp. $\mathfrak{p}^2$); in particular, if $e_p$ is the element of $1 + \mathfrak{p}$ such that $\log e_p = p$ (resp. $\log e_2 = 4$) then the isomorphism of $\mathbf{Z}_p$ onto $1 + \mathfrak{p}$, which is inverse to $x \mapsto \frac{1}{p} \log x$, (resp. $x \mapsto \frac{1}{4} \log x$) is $y \mapsto e_p^y$ (cf. Gen. Top., III, p. 323, exerc. 25).

e) Show that for every $a \in 1 + \mathfrak{p}$ the continuous function $x \mapsto a^x$, defined on $\mathbf{Z}_p$, admits a derivative equal to $a^x \log a$ at every point; deduce from this that the function $\log x$ admits a derivative equal to $1/x$ at every point of $1 + \mathfrak{p}$.