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The question is fairly straightforward:

Is there a topological division ring $D$ of characteristic $p$ that is Hausdorff and connected?

I do not assume that inversion is continuous, that is, only addition and multiplication are assumed to be continuous under the topology in question.

Since the connected component of $0$ is an ideal, any topological division ring is either connected or totally disconnected. I cannot think of a single connected topological $\mathbb{F}_p$ algebra, but couldn't point down a proof that they are all totally disconnected either. I'm asking this here as I suppose this might be known.

Henrique Augusto Souza
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    related question gives many examples of metric and path connected division algebras of arbitrary positive characteristic: https://projecteuclid.org/journalArticle/Download?urlId=10.1215%2Fkjm%2F1250524535 – Henrique Augusto Souza Feb 06 '24 at 14:33
  • Does the previous comment not completely answer your question? – Eric Wofsey Feb 06 '24 at 16:49
  • @EricWofsey it does, but I'm unsure what is the procedure... do I delete the question? Leave it here so it might show up on google search, since the related question is not quite the same? (https://math.stackexchange.com/questions/1523780/connected-field-must-be-path-connected?rq=1) – Henrique Augusto Souza Feb 06 '24 at 16:58
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    You can post an answer here with the link to that paper and summarize its results. – Eric Wofsey Feb 06 '24 at 17:00

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In the paper:

Alan G. Waterman, George M. Bergman, Connected fields of arbitrary characteristic, J. Math. Kyoto. Univ. 5-2, 1966. (linked),

the authors present for every characteristic a construction of a topological division ring $D$ of that characteristic that is Hausdorff and connected. Moreover, $D$ is actually commutative with continuous inversion (so a topological field) and path connected.

The precise construction is as follows: start with any integral domain $k$ and let $R$ be a polynomial ring over $k$ where one independent variable $T_\alpha$ is added to $k$ for every real $\alpha \in [0,1]$, and then kill the relations $T_0 = 0$, $T_1 = 1$. The authors define a norm $|\cdot|$ on $R$ using the norm on $(0,1)$ which induces a ring topology such that the map $(0,1) \to R$ that maps $\alpha$ to $T_\alpha$ is an isometry, from which follows that this ring is path connected. The norm is such that $|T_\alpha - T_\beta| = |\alpha - \beta|$.

This topology is later show to extend into a Hausdorff field topology on the field of fractions $K$ of $R$ such that the inclusion $R \to K$ is a homeomorphism onto its image. In particular, since $0$ and $1$ are connected by a path in $R$, they so are in $K$, which shows that $K$ is also path connected. It is clear that $K$ has the same characteristic as the initial integral domain $k$.

Henrique Augusto Souza
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