Exercise 3, § 2

Every $p$-adic number $x$ (Exercise 2) can be written in the form

where the $\alpha_k$ are rational integers, those of index $k < 0$ being zero for all but a finite number of values of the index $k$. Moreover, if

the rational numbers $\sum_{k=-\infty}^{-1} \alpha_k p^k$ and $\sum_{k=-\infty}^{-1} \beta_k p^k$ are congruent mod 1. Let $\varphi_p(x)$ be the class mod 1 of the rational number $\sum_{k=-\infty}^{-1} \alpha_k p^k$.

a) Show that $\varphi_p$ is a continuous homomorphism of the topological group $\mathbf{Q}_p$ into $\mathbf{T}$, and that for every continuous homomorphism $f$ of $\mathbf{Q}_p$ into $\mathbf{T}$ there exists a unique $a \in \mathbf{Q}_p$ such that $f(x) = \varphi_p(ax)$ for all $x \in \mathbf{Q}_p$. [Note that the knowledge of the elements $f(p^k)$ for $k \leq 0$ determines $f$ uniquely; show that each of these elements is the class mod 1 of a fraction whose denominator is a power of $p$; hence show that there exists $a \in \mathbf{Q}_p$ such that $f(p^k) = \varphi_p(ap^k)$ for all $k \leq 0$.]

b) Show that, if $\varphi : \mathbf{R} \to \mathbf{T}$ is the canonical homomorphism, then $\varphi(x) = \sum_p \varphi_p(x)$ for all $x \in \mathbf{Q}$, the sum being over all prime numbers $p$ (observe that all but a finite number of terms of this sum will be zero).