Exercise 25, § 6

a) Let $a \neq 0$ be a $p$-adic number. Then the mapping $n \to a^n$ is continuous on $\mathbf{Z}$ (as a subspace of $\mathbf{Q}_p$) if and only if $a \in 1 + \mathfrak{p}$. If this condition is satisfied, show that $n \to a^n$ is uniformly continuous on $\mathbf{Z}$ and extends to a continuous homomorphism of $\mathbf{Z}_p$ into $U$, denoted by $x \to a^x$, which is injective if $a \neq 1$. If $p > 2$ and if $m$ is the largest integer such that $a \in 1 + \mathfrak{p}^m$, then $x \to a^x$ is an isomorphism of $\mathbf{Z}_p$ onto $1 + \mathfrak{p}^m$. How must this statement be modified when $p = 2$?

b) Show that every continuous homomorphism of $\mathbf{Z}_p$ into $U$ is of the form $x \to a^x$ for some $a \in 1 + \mathfrak{p}$ [if $f$ is such a homomorphism and $f(1) = a$, then $f(n) = a^n$ for $n \in \mathbf{Z}$].

c) Show that the mapping $(x, y) \to x^y$ is continuous on $(1 + \mathfrak{p}) \times \mathbf{Z}_p$. If $p > 2$ and $b \in \mathbf{Z}_p$, and if $m$ is the largest integer such that $b \in \mathfrak{p}^m$, then the mapping $x \to x^b$ is an isomorphism of the multiplicative group $1 + \mathfrak{p}$ onto the multiplicative subgroup $1 + \mathfrak{p}^{m+1}$ [use Exercise 24 b)]. How must this statement be modified when $p = 2$?

d) If $n$ is an integer prime to both $p - 1$ and $p$, show that the mapping $x = x^n$ is an automorphism of the multiplicative group $U$ [use Exercise 24 c)].