Proposition 1

If $K$ is a locally compactfield, thefunction mod, belongs to $\mathscr{V}(K)$ (§ 6, no. 1). Moreover: (i) If $s > 0$ is such that $(\operatorname{mod}_K)^s = g$ is an absolute value, then $g$ defines the topology on $K$. (ii) If $K$ is not discrete and mod, is an ultrametric absolute value, there exists a normed discrete valuation $v$ on $K$ whose ring is compact and whose residuefield is finite with $q$ elements, so that $\operatorname{mod}, = q^{-v}$. The topology on $K$ is defined by $v$.

This follows from § 6, no. 1, Proposition 1, § 5, no. 1, Proposition 2 and Integration, Chapter VII, § 1, no. 10, Propositions 12 and 13.