Two notes on valued fields

Abstract: This paper studies two questions on valued fields: the metric dimension induced by an absolute value, and the uniform openness of multiplication. For nontrivial non-archimedean absolute values, we prove that the metric dimension equals the density character. In the archimedean case, it is 2 for subfields of $\mathbb{R}$, while for non-real subfields of $\mathbb{C}$ it is either 2 or 3, depending on invariance under complex conjugation. We also show that multiplication is uniformly open on every valued field. Finally, we prove that this property is genuinely metric, not purely topological, even on $\mathbb{R}$ with a suitable compatible choice of metric.