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Using AC one may prove that there are $2^{\mathfrak{c}}$ field automorphisms of the field $\mathbb{C}$. Certainly, only the identity map is $\mathbb{C}$-linear ($\mathbb{C}$-homogenous) among them but are all these automorphisms $\mathbb{R}$-linear?

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    perhaps this mathoverflow question will interest you http://mathoverflow.net/questions/24047/ultrafilters-and-automorphisms-of-the-complex-field2012-06-18

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An automorphism of $\mathbb C$ must take $i$ into $i$ or $-i$. Thus an automorphism that is $\mathbb R$-linear must be the identity or conjugation.

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    Is that mean that all aotumorphisms on C fix R?2013-05-06