Ostrowski's Theorem for topological rings? 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.
 A: Thanks to YCor's examples in the comments, I decided this question was worth a deep dive. It turns out on the one hand that

There are lots of exotic (Hausdorff) field topologies on $\mathbb Q$.

But on the other hand, it turns out that

Every locally compact (Hausdorff) ring topology on a field is induced by an absolute value.

See Theorem 16.3 In Warner's Topological Rings. Shanks and Warner also showed that every locally bounded (Hausdorff) ring topology on $\mathbb Q$ comes from an absolute value. Here a topological ring $R$ is locally bounded if there is a neighborhood $B$ of 0 which is bounded in the sense that for every neighborhood $U$ of 0 there is a neighborhood $V$ of 0 such that $VB \subseteq U$. A partial extension to global fields was given by Nichols and Cohen (not in alphabetical order).
It seems there's been work on constructing exotic topologies on more general rings and fields, but I didn't come across positive results constraining these more general topologies under reasonable conditions.
A: The following relevant classification result, due to Kowalsky and Dürbaum [2], appears in Appendix B of Engler and Prestel [1].
Let $(K,\tau)$ be a topological field. Then $\tau$ is called a V-topology if for every neighbourhood $W\ni0$, there exists a neighbourhood $U\ni0$ such that $(K\smallsetminus W)(K\smallsetminus W)\subseteq K\smallsetminus U$ (that is, for any $x,y\in K$, if $xy\in U$, then $x\in W$ or $y\in W$).

Theorem: The V-topologies on a given field are exactly the topologies induced by valuations (with arbitrary value groups) or by archimedean absolute values.

Note that nonarchimedean absolute values are also covered, being special cases of valuations (with, confusingly enough, value groups that are archimedean, i.e., rank 1).
In the special case of global fields (including number fields), all valuations have rank 1, i.e., they are equivalent to nonarchimedean absolute values (e.g., see Thm. 2.1.4 and Cor. 3.2.5 in [1]). Thus:

Corollary: If $K$ is a global field, the topologies on $K$ induced by absolute values are exactly the V-topologies.

References:
[1] Antonio J. Engler and Alexander Prestel, Valued fields, Springer, 2005.
[2] Hans-Joachim Kowalsky and Hansjürgen Dürbaum, Arithmetische Kennzeichnung von Körpertopologien, Journal für die reine und angewandte Mathematik 191 (1953), 135­–152.
