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November 16th, 2006, 08:14 AM   #1
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Pseudo-finite fields

How to show that pseudo-finite fields are the infinite models of the theory of finite fields?
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March 5th, 2009, 03:20 PM   #2
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Re: Pseudo-finite fields

Pseudo-finite fields are by definition infinite models of the theory of finite fields. One can take a field theoretic definition of pseudo-finite fields as well. It goes as follows: A field is pseudo-finite if it is perfect, pseudo algebraically closed and it has a unique extension of degree n for every n>0. Of course the first and the second definitions are equivalent. To show that a field which is pseudo-finite (i.e perfect, pseudo algebraically closed and it has a unique extension of degree n for every n>0) is an infinite model of the theory of finite fields one has to show that all these three properties are first order expressible. This is not trivial but can be done. Then one shows that these first order statements corresponding to the three properties are satisfied by all large enough finite fields, i.e. they hold in the theory of finite fields. Finally we have to show that a field satisfying the three properties has to be infinite. This comes from pseudo algebraically closedness. Finite fields are not pseudo algebraically closed, hence the only fields which satisfy the three properties are the infinite models of the theory of finite fields. (Which do exist because of the compactness, the theory of finite fields have arbitrarily large models, therefore they have infinite models as well.)
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