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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.

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

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  • $\begingroup$ Note that $K$ must be uncountable. $\endgroup$ – Pedro Tamaroff Dec 24 '14 at 19:22
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    $\begingroup$ And every element of $\mathbb R\setminus K$ must be algebraic. $\endgroup$ – Thomas Andrews Dec 24 '14 at 19:23
  • $\begingroup$ Looks like you'd need to adjoin a set of positive measure to $\mathbb{Q}$ as a basis $\endgroup$ – David Peterson Dec 24 '14 at 19:23
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    $\begingroup$ Is there any elemantary proof for nonexistence ? $\endgroup$ – mesel Dec 24 '14 at 19:30
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    $\begingroup$ 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. $\endgroup$ – Lubin Dec 24 '14 at 20:47
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The Artin-Schreier theorem implies that $\;[\Bbb C:K]\le2\;$ , and from here that the answer is no .

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