Exercise 13, § 3

Let $K$ be a non-discrete Hausdorff topological division ring. Let $R$ be the set of all $x \in K$ such that $\lim_{n \to \infty} x^n = 0$, and let $N$ be the complement of the set $R \cup R^{-1}$. Show that there exists an absolute value on $K$ which defines the topology of $K$ if and only if (i) $R$ is open in $K$; (ii) for each neighbourhood $V$ of $0$ in $K$, there is a neighbourhood $U$ of $0$ in $K$ such that $R.U \subset V$; and (iii) if $x \in R$ and $y \in R \cup N$, then $yx \in R$.

To show that these conditions are sufficient, prove successively that: a) $N$ is a normal subgroup of the multiplicative group $K^*$ of non-zero elements of $K$. b) In the quotient group $K^*/N$, put $\dot{x} \leq \dot{y}$ if there exist $x \in \dot{x}$ and $y \in \dot{y}$ such that $yx^{-1} \in R \cup N$; show that this relation is an ordering compatible with the group structure of $K^*/N$, and that the ordered group so defined is isomorphic to a subgroup of the additive group $\mathbf{R}$ (use Exercise 1 of Chapter V, § 3). c) Complete the proof with the help of Exercise 12. In particular, the topology of a non-discrete locally compact topological field can be defined by an absolute value.