Exercise 12, § 3

Let $K$ be a non-discrete, Hausdorff, topological division ring, and let $\varphi : K \to \mathbf{R}_+$ be a mapping such that $\varphi(0) = 0, \varphi(xy) = \varphi(x)\varphi(y)$ and such that if $V_n$ denotes the set of all $x \in K$ such that $\varphi(x) \leq 1/n$, the sets $V_n$ form a fundamental system of neighbourhoods of $0$ in $K$.

a) Show that there exists a real number $a > 0$ such that, for all $x \in K$,

[if not, there would exist a sequence $(x_n)$ of points of $K$ such that both $(1 + x_n)^{-1}$ and $x_n(1 + x_n)^{-1}$ tended to zero]. Hence show that if $\psi(x) = (\varphi(x))^{\alpha}$ we have for $\alpha$ sufficiently small.

b) Show that if $n = 2^p$ we have $\psi \left( \sum_{i=1}^n x_i \right) \leq n \sup (\psi(x_i))$, and deduce that for every integer $m > 0$ we have $\psi(m) \leq 2m$.

c) Deduce from b) that for every $n = 2^p$ and every $x \in K$ we have $\psi((1 + x)^{n-1}) \leq 2n(1 + \psi(x))^{n-1}$, and hence that $\psi$ is an absolute value on $K$ which defines the topology of $K$.