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What is a $\mathbb{Q}$-automorphisms of $\mathbb{R}$ ? Does this mean, I have to think if $\mathbb{R}$ as a $\mathbb{Q}$ vector space ? Or does that mean, that it is an automorphism on $\mathbb{R}$ that restricted to $\mathbb{Q}$ is the identity (this was some definition I picked up on the web, searching for an explanation)?

And can someone please tell me how I could find all of these automorphisms ?

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What is the context? – Julian Kuelshammer Nov 6 '12 at 22:29
up vote 2 down vote accepted

It's probably a field automorphism of $\mathbb R$. Any such automorphism automatically is the identity when restricted to $\mathbb Q$.

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Amy field automorphism of $\mathbb R$ is the identity on all of $\mathbb R$. – GEdgar Nov 6 '12 at 22:52
@GEdgar: *continuous – Kris Joanidis Nov 6 '12 at 22:53
Even if no hypothesis of continuity is assumed. – GEdgar Nov 6 '12 at 22:56
@GEdgar, you're right of course. But that does not make what I said false... – lhf Nov 6 '12 at 23:44
Regarding the question of how to find such automorphisms. there is the identity automorphism. There are no nontrivial such automorphisms that are continuous (rather easy) or measurable (challenging exercise). The existence of a non-trivial automorphism depends on the axiom of choice. – Lior B-S Nov 7 '12 at 20:06

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