Is it true that the only proper subfield of $\mathbb C$ , which is path connected as a subset of $\mathbb C$ , must be $\mathbb R$ ?
NOTE : The answer to this Subfields of $\mathbb{C}$ which are connected with induced topology indicates the claim stated is true . But the answer there only gives a reference and no proof ( or outline of proof) of the result , and the paper cited is highly technical . I would like to see if there is any elementary ( at least more elementary than the cited paper if possible ) proof .