Proposition 6

The completion $\hat{K}$ of a division ring $K$ endowed with an absolute value $|x|$ is a division ring, and the function $|x|$ can be extended by continuity to an absolute value on $\hat{K}$, which defines the topology of $\hat{K}$.

Let $\mathfrak{F}$ be a Cauchy filter on $K$ (with respect to the additive uniformity) which does not have $0$ as a cluster point; to show that $\hat{K}$ is a division ring it is enough to establish that the image of $\mathfrak{F}$ under the mapping $x \to x^{-1}$ is a Cauchy filter base (Chapter III, § 6, no. 8, Proposition 7). Now, by hypothesis there exists a real number $\alpha > 0$ and a set $A \in \mathfrak{F}$ such that $|x| \geq \alpha$ for all $x \in A$; on the other hand, for each $\varepsilon > 0$ there exists a set $B \in \mathfrak{F}$ such that $B \subset A$ and $|x - y| \leq \varepsilon$ for all $x \in B$.

and $y \in B$; hence

and the first part of the proposition follows. The invariant metric $|x - y| = d(x, y)$ extends by continuity to a metric on $\hat{K}$ (§ 2, no. 1, Proposition 1) which defines the topology of $\hat{K}$ and is invariant by the principle of extension of identities; we continue to denote this invariant metric by $d(x, y)$. If we put $|x| = d(0, x)$ for $x \in \hat{K}$, it is clear that $|x|$ is the extension by continuity of the function $|x|$ on $K$ and is therefore an absolute value on $\hat{K}$ by the principle of extension of identities.