Exotic subfield of R

We build here an example of strict subfield indénombrable of $\ mathbb {R}$ using the Lemme of Zorn (and thus of the Axiome of the choice).

That is to say E the whole of the subfields of $\ mathbb {R}$ not containing $\ sqrt {2}$ . E nonempty (because it contains for example $\ mathbb {Q}$ ) and is ordered (partially) by inclusion. It is checked easily that it is then an inductive Ensemble. According to the lemma of Zorn it thus has a maximum element K. The maximality of K makes it possible to show that the extension $K \ to \ mathbb {R}$ is algebraic; the extension $K \ to \ mathbb {R}$ is thus also, which involves that K is indénombrable. Lastly, K is a strict subfield of $\ mathbb {R}$ because it does not contain $\ sqrt {2}$ . Let us note that $K$ is strictly included dans $\ mathbb {R}$ : in the contrary case, the automorphism of body of $K$ fixing the elements of K and sending $\ sqrt {2}$ on $- \ sqrt {2}$ would be an automorphism of body of $\ mathbb {R}$ other than the identity, which is absurd.

See too

Automorphisms of noncontinuous bodies of C

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