# Automorphism groups of fields

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
General field automorphisms. And indeed, |.| = cardinality. –  THC May 10 '12 at 15:25
@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
The fields ${\mathbb{Q}}_p$ have only the identity automorphism, like the reals. –  Lubin May 10 '12 at 16:31
This question is relevant: mathoverflow.net/questions/22897/… –  Kevin Ventullo May 10 '12 at 17:56