Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property:

every maximal ideal of this ring is the subset of all functions vanishing at a common point.

If we follow the argument in the proof of this property, it can be seen that $\mathbb{R}$ can be replaced by $\mathbb{C}$ and still similar property holds for new ring (we have to slightly modify argument).

What if we place $\mathbb{Q}$? Here $\mathbb{Q}$ is disconnected with induced topology from $\mathbb{C}$ and so the only continuous function from $[0,1]$ to $\mathbb{Q}$ are constant (am I correct?). So, the property no longer holds true with replacement of $\mathbb{R}$ by $\mathbb{Q}$.

Question: Are there subfields of $\mathbb{C}$ other than $\mathbb{R}$ which are connected as topological spaces with induced topology from $\mathbb{C}$?

Solutions Collecting From Web of "Subfields of $\mathbb{C}$ which are connected with induced topology"

Yes, there are proper connected subfields of $\mathbb{C}$ other than $\mathbb{R}$.

A construction can be found in a paper by J.Dieudonné:

J.Dieudonné, Sur les corps topologiques connexes, C.R. Acad. Sci. Paris vol. 221 (1945) pp. 396-398.

I don’t have access to the above paper at the moment.

It can in fact be proved that the only proper path-connected subfield of $\mathbb{C}$ is $\mathbb{R}$, see a more detailed discussion at:

Niel Shell, Connected and disconnected fields, Topology and its Applications, (27) 1987, 37-50.

Also, since the only maximal ideal of $\mathbb{Q}$ is $\{0\}$, the statement about maximal ideals remains true.