Ostrowski's theorem classifies all absolute values on a number field $K$.

**Questions:**

More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?

In particular, is every Hausdorff topology on a number field $K$ making $K$ into a topological field induced by an absolute value?

It would already be interesting to understand this when $K= \mathbb Q$. On the other hand, I'd be interested to understand this question for more general fields and rings as well. For "large" fields / rings, I imagine one might need to consider valuations in more general value groups as well. But I don't know a generally-accepted definition of "archimedean valuation" not over $\mathbb R$, so I'm not quite sure how to formulate a potentially-correct statement saying that "every topology comes from a generalized absolute value" in this context.