A proper subfield of $\mathbb{R}$ must not be connected?

Question

Let $\mathbb{R}$ denote the real line as a topological field, if $F$ is a proper subfield of $\mathbb{R}$, then $F$ must not be connected?

Answer 1

Let $x \in \mathbb{R}\setminus{F}$. We have $F = (F \cap (-\infty , x)) \cup (F \cap (x,\infty))$ and as $F$ contains $\mathbb{Z}$ the sets $F \cap (-\infty , x)$ and $F \cap (x,\infty)$ are not empty.

Answer 2

A connected subset of $\mathbb{R}$ is an interval. If $F$ is a subfield of $\mathbb R$ then it must contain $\mathbb Q$ and hence to be connected it must be all of $\mathbb R$