Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.

Question

Is there a field $K \subset \mathbb{R}$ such that $1 < [\mathbb{R} : K] < \infty$? i.e a proper subfield of $\mathbb{R}$ such that the field extension $\mathbb{R}/K$ is finite.

Answer 1

The Artin-Schreier theorem implies that $\;[\Bbb C:K]\le2\;$ , and from here that the answer is no .