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Hi there,

is there a classification/characterization of fields K for which the automorphism group Aut(K) has the property that |Aut(K)| < |K| (e.g. finite fields, the rationals and reals) ? What about the same question for real-closed fields K ? Many thanks ...

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What automorphisms? Do you mean $| \cdotp |$ =cardinaity? – Marc Palm May 10 '12 at 14:51
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@Gerhard: Unfortunately, the reals have lots of vector-space automorphisms over the prime field $\mathbb Q$ but no nontrivial field automorphisms. (I like to see things turn into set theory, but this one will need more work.) – Andreas Blass May 10 '12 at 16:03
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The fields ${\mathbb{Q}}_p$ have only the identity automorphism, like the reals. – Lubin May 10 '12 at 16:31
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This question is relevant: mathoverflow.net/questions/22897/… – Kevin Ventullo May 10 '12 at 17:56
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Gerhard, I guess we are using the terminology "prime field" with different meanings. To me, it means a field with no proper subfields, i.e., either the rationals or a finite field of prime order. – Andreas Blass May 11 '12 at 18:46

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