Automorphism of noncontinuous body of C

Only the Automorphisme of body of $\ mathbb {R}$ is l'identité and that the only automorphisms of continuous bodies of $\ mathbb {C}$ are l'identité and the conjugaison. The use of the Axiome of the choice (twice) makes it possible to build other automorphisms of body of $\ mathbb {C}$ .

That is to say E the whole of the Subfield of $\ mathbb {C}$ 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 Maximalité of K makes it possible to show that the extension $K (\ sqrt {2}) \ to \ mathbb {C}$ is algebraic and $\ mathbb {C}$ is algebraically closed; any automorphism of body of $K (\ sqrt {2})$ is thus prolonged in an automorphism of body of $\ mathbb {C}$ (this result is traditional and uses to him also the Axiome of the choice). By considering the automorphism of $K (\ sqrt {2})$ fixing K point by point and sending $\ sqrt {2}$ on $- \ sqrt {2}$ , one then obtains an automorphism of body of $\ mathbb {C}$ other than identity and the conjugation: it is thus not continuous and even discontinuous in any point. One can then show that it is not measurable and that the image of $\ mathbb {R}$ is dense: thus, the axiom of the choice involves the existence of a dense subfield of $\ mathbb {C}$ isomorphous with $\ mathbb {R}$ .

See too

exotic Subfield of R

Random links: