Are there other topological field structures on the complex numbers which make the inclusion of the reals a topological embedding?

Question

We consider the field of reals $\mathbb{R}$ with its standard topological field structure.

Let $F = \mathbb{R}[t]/(t^2 + 1)$, as a field extension of $\mathbb{R}$. We know this extension is isomorphic to $\mathbb{C} \supset \mathbb{R}$ via sending the class of $t$ to $\pm i$.

I was wondering, does there exist a topology $\tau$ on $F$ for which:

1. $F$ is a topological field, and

2. the field inclusion $\mathbb{R} \hookrightarrow F$ is a topological embedding;

3. but, the aforementioned field isomorphism $F \rightarrow \mathbb{C}$ (over $\mathbb{R}$) is not a homeomorphism? (Here $\mathbb{C}$ is given its usual topology as the complex plane.)

I'm not quite sure how to approach this question. Would anyone have any suggestions?

Answer 1

If $F$ is a topological field that contains $\mathbb{R}$ as a topological subfield, then in particular $F$ is a topological vector space over $\mathbb{R}$. The topology on a finite-dimensional topological vector space over $\mathbb{R}$ is uniquely determined by the vector space structure (see How to endow topology on a finite dimensional topological vector space?). In particular, the only topology that makes $\mathbb{C}$ a topological field and restricts to the usual topology on $\mathbb{R}$ is the usual topology on $\mathbb{C}$.