Exercise 11, § 3

Let K be a separated non-discrete topological field, $\varphi$ a mapping from K into $\mathbf{R}_+$, such that $\varphi(0) = 0$, $\varphi(xy) = \varphi(x)\varphi(y)$, and that, if $V_n$ denotes the set of $x \in K$ such that $\varphi(x) \leq 1/n$, the $V_n$ form a fundamental system of neighbourhoods of 0 in K.

a) Now prove that there exists a number $a > 0$ such that one has $\varphi(1 + x) \leq a(1 + \varphi(x))$ (otherwise, there would exist a sequence $(x_n)$ of points of K such that $(1 + x_n)^{-1}$ and $x_n(1 + x_n)^{-1}$ both tend to 0). Deduce that if one sets $\psi(x) = (\varphi(x))^{\alpha}$, one has

for $\alpha$ sufficiently small.

b) Now prove that if $n = 2^p$, one has $\psi \left( \sum_{i=1}^n x_i \right) \leq n \sup (\psi(x_i))$; deduce that for every integer $m > 0$, one has $\psi(m) \leq 2m$.

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