Exercise 17, § 1

Let $E$ be the product $(\mathbf{Q}_p)^n$ of $n$ factors identical to the field $\mathbf{Q}_p$ of $p$-adic numbers (III, p. 84, exerc. 23), endowed with the product topological group structure of the topological additive group structures of its factors, and with its vector space structure of dimension $n$ over the field $\mathbf{Q}_p$. If $v_p$ denotes the additive $p$-adic valuation on $\mathbf{Q}_p$, one puts $|x|_p = p^{-v_p(x)}$ for every $x \in \mathbf{Q}_p$ and $\neq 0$, $|0|_p = 0$; for every $\mathbf{x} = (x_i)_{1 \leq i \leq n}$ in $E$, one puts $||\mathbf{x}|| = \sup_i |x_i|_p$ (cf. IX, § 3, nos 2 and 3).

a) For a subset A of E to be relatively compact, it is necessary and sufficient that $\sup_{x \in A} \|x\| < +\infty$. (Remark that $v_p$ is a continuous mapping from $\mathbf{Q}_p$ into $\mathbf{Z}$ endowed with the discrete topology.)

b) If G is a closed subgroup of E, G is a topological module over the ring $\mathbf{Z}_p$ of $p$-adic integers (if $x \in G$, one has $nx \in G$ for every $n \in \mathbf{Z}$, and $\mathbf{Z}$ is dense in $\mathbf{Z}_p$).

c) If K is a compact subgroup of E, there exists a free system of $m \leq n$ points $a_i$ ($1 \leq i \leq m$) of E, such that K is the direct sum of the m groups $\mathbf{Z}_p . a_i$. (Use a) to show that, if $(e_j)_{1 \leq j \leq n}$ is the canonical basis of E, there is an integer $k \in \mathbf{Z}$ such that K is contained in the direct sum of the n groups $\mathbf{Z}_p . p^k e_j$; then apply the theory of modules over a principal ideal domain (A, VII, § 4, no 2).)

d) If G is a closed non-compact subgroup of E, G contains a vector subspace of dimension 1, $\mathbf{Q}_p . a$ with $a \neq 0$. (Let C be the compact subset of E formed by the points x such that $\|x\| = 1$; now prove that $G \cap C$ contains a sequence of points $(x_r)_{r \in \mathbf{N}}$ such that $p^{-r} x_r \in G$, and take for a a value of adherence of this sequence).

e) Let G be a closed subgroup of E, V the largest vector subspace contained in G, W a supplementary subspace of V; now prove that $W \cap G$ is compact and that G is the direct sum of V and $W \cap G$ (reason as in th. 2 of VII, p. 5).

f) Given a subgroup G of E, one denotes by $G^*$ the set of points $u = (u_i)$ of E such that $(u | x) = \sum_{i=1}^n u_i x_i$ belongs to $\mathbf{Z}_p$ for every $x = (x_i) \in G$. Now prove that $G^*$ is closed in E, that $(\overline{G})^* = G^*$ and that $(G^*)^* = \overline{G}$.

g) State and exercise prove the analogues of exerc. 2 to 8 for the closed subgroups of E. In particular, every closed subgroup and every separated quotient group of a group of the form $(\mathbf{Q}_p)^r \times (\mathbf{Q}_p/\mathbf{Z}_p)^s \times (\mathbf{Z}_p)^t \times F$ (where F is the product of a finite number of cyclic groups whose order is a power of p) is a group of the same form.