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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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R linearity would force the aut. to preserve R. Hence the only ones are the identity and reflection on the i axis. – Michael Luo Jun 18 '12 at 12:38
perhaps this mathoverflow question will interest you… – Holdsworth88 Jun 18 '12 at 12:41
up vote 3 down vote accepted

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? – safellh May 6 '13 at 0:17

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