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

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.

## marked as duplicate by Watson, C. Falcon, Shailesh, user91500, JMPFeb 1 '17 at 9:41

• Note that $K$ must be uncountable. – Pedro Tamaroff Dec 24 '14 at 19:22
• And every element of $\mathbb R\setminus K$ must be algebraic. – Thomas Andrews Dec 24 '14 at 19:23
• Looks like you'd need to adjoin a set of positive measure to $\mathbb{Q}$ as a basis – David Peterson Dec 24 '14 at 19:23
• A very natural question, but as @Timbuc’s answer points out, Artin-Schreier forbids such a field. Do you know the proof that $\mathbb R$ has no nontrivial automorphisms? It’s much more basic than A-S, and you can prove it yourself. It demonstrates the weaker proposition that there are no fields $K$ over which $\mathbb R$ is normal. – Lubin Dec 24 '14 at 20:47
The Artin-Schreier theorem implies that $\;[\Bbb C:K]\le2\;$ , and from here that the answer is no .