Exercise 23, § 6

If $p$ is any prime number and $x$ is any non-zero rational number, let $v_p(x)$ denote the exponent of $p$ in the decomposition of $x$ into prime factors. The mapping $v_p$ of $\mathbf{Q}^*$ into $\mathbf{Z}$ is called the $p$-adic valuation on $\mathbf{Q}$. The set of all $x \in \mathbf{Q}$ such that either $x = 0$ or $v_p(x) \geq m$ (for some $m \in \mathbf{Z}$) is the fractional ideal $(p^m)$ of $\mathbf{Q}$.

a) Show that the ideals $(p^m)\ (m \in \mathbf{Z})$ form a fundamental system of neighbourhoods of o in $\mathbf{Q}$ for a locally retrobounded topology (Exercise 22) $\mathfrak{T}_p$ on $\mathbf{Q}$, called the $p$-adic topology. The completion $\mathbf{Q}_p$ of $\mathbf{Q}$ in this topology is a field called the $p$-adic field, whose elements are called $p$-adic numbers. The closure $\mathbf{Z}_p$ of $\mathbf{Z}$ in $\mathbf{Q}_p$ is a compact open principal ideal domain in $\mathbf{Q}_p$, in which $\mathfrak{p} = p\mathbf{Z}_p$ is the only non-zero prime ideal; $\mathbf{Z}_p/\mathfrak{p}$ is isomorphic to the prime field $\mathbf{F}_p = \mathbf{Z}/(p)$, and more generally the additive quotient group $\mathfrak{p}^m/\mathfrak{p}^n$ is isomorphic to

The elements of $\mathbf{Z}_p$ are called $p$-adic integers.

b) Show that every continuous homomorphism of the additive group $\mathbf{Q}_p$ into itself is of the form $x \to ax$, where $a \in \mathbf{Q}_p$ [if $f$ is such a homomorphism, show that $f(rx) = rf(x)$ for all $r \in \mathbf{Q}$, and then pass to the limit in $\mathbf{Q}_p$].

c) Show that every compact subgroup $G \neq \{0\}$ of the additive group $\mathbf{Q}_p$ is identical with one of the $\mathfrak{p}^n$ ($n \in \mathbf{Z}$), and that there is no non-compact closed subgroup of the additive group $\mathbf{Q}_p$, except $\mathbf{Q}_p$ itself. [Let $m$ be the largest integer such that $G \subset \mathfrak{p}^m$; by considering the quotient group $(G + \mathfrak{p}^n)/\mathfrak{p}^n$ for $n > m$, show that $G + \mathfrak{p}^n = \mathfrak{p}^m$, and then use formula (1) of § 3, no. 1].